Jukes Cantor model with root distribution $\pi = \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right)$.
The transition matrix associated with edge $i$ is of the form
$$M_j = \begin{pmatrix}
a_i & b_i & b_i & b_i\\
b_i & a_i & b_i & b_i\\
b_i & b_i & a_i & b_i\\
b_i & b_i & b_i & a_i
\end{pmatrix}$$ and the Fourier parameters are $\left(x^{(i)}_i, x^{(i)}_2, x^{(i)}_2, x^{(i)}_2\right)$ for all edges $i$.
Phylogenetic tree 4-0-1
Evolutionary model
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Summary
Dimension: 6
Degree: 34
Probability coordinates: 15
Fourier coordinates: 13
dim of Singular locus: -
deg of Singular locus: -
MLdeg: 17332
EDdeg: 290
Degree: 34
Probability coordinates: 15
Fourier coordinates: 13
dim of Singular locus: -
deg of Singular locus: -
MLdeg: 17332
EDdeg: 290
Probability parametrization
$\begin{align}
p_{\texttt{ACAC}} &= \frac{1}{2}(a_1a_3a_5b_2b_4 + a_1a_4b_2b_3b_5 + 2a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4a_5b_1b_3 + 2a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 2b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACAA}} &= \frac{1}{2}(a_1a_3a_4a_5b_2 + 3a_1b_2b_3b_4b_5 + a_2a_3a_4b_1b_5 + a_2a_5b_1b_3b_4 + 2a_2b_1b_3b_4b_5 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACCG}} &= \frac{1}{2}(a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4 + a_1b_2b_3b_4b_5 + a_2a_3a_5b_1b_4 + a_2a_4b_1b_3b_5 + 2a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACGC}} &= \frac{1}{2}(a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4 + a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4a_5b_1b_3 + 2a_2b_1b_3b_4b_5 + a_3a_5b_1b_2b_4 + a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{AACC}} &= \frac{1}{2}(a_1a_2a_3a_4b_5 + a_1a_2a_5b_3b_4 + 2a_1a_2b_3b_4b_5 + a_3a_4a_5b_1b_2 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4 + 7b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACGA}} &= \frac{1}{2}(a_1a_3b_2b_4b_5 + a_1a_4a_5b_2b_3 + 2a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 + a_2b_1b_3b_4b_5 + a_3a_5b_1b_2b_4 + a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{AACA}} &= \frac{1}{2}(a_1a_2a_3b_4b_5 + a_1a_2a_4a_5b_3 + 2a_1a_2b_3b_4b_5 + a_3a_5b_1b_2b_4 + 2a_3b_1b_2b_4b_5 + 3a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACAG}} &= \frac{1}{2}(a_1a_3a_5b_2b_4 + a_1a_4b_2b_3b_5 + 2a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 + a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 3b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{AAAC}} &= \frac{1}{2}(a_1a_2a_3a_5b_4 + a_1a_2a_4b_3b_5 + 2a_1a_2b_3b_4b_5 + 3a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + 2a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACGG}} &= \frac{1}{2}(a_1a_3a_4b_2b_5 + a_1a_5b_2b_3b_4 + 2a_1b_2b_3b_4b_5 + a_2a_3a_4b_1b_5 + a_2a_5b_1b_3b_4 + 2a_2b_1b_3b_4b_5 + a_3a_4a_5b_1b_2 + a_3a_4b_1b_2b_5 + a_5b_1b_2b_3b_4 + 5b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{AACG}} &= \frac{1}{2}(a_1a_2a_3b_4b_5 + a_1a_2a_4b_3b_5 + a_1a_2a_5b_3b_4 + a_1a_2b_3b_4b_5 + a_3a_5b_1b_2b_4 + 2a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + 2a_4b_1b_2b_3b_5 + a_5b_1b_2b_3b_4 + 5b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{ACCC}} &= \frac{1}{2}(a_1a_3a_4b_2b_5 + a_1a_5b_2b_3b_4 + 2a_1b_2b_3b_4b_5 + a_2a_3a_4a_5b_1 + 3a_2b_1b_3b_4b_5 + 2a_3a_4b_1b_2b_5 + 2a_5b_1b_2b_3b_4 + 4b_1b_2b_3b_4b_5) \\[0.5em]
p_{\texttt{AAAA}} &= \frac{1}{2}(a_1a_2a_3a_4a_5 + 3a_1a_2b_3b_4b_5 + 3a_3a_4b_1b_2b_5 + 3a_5b_1b_2b_3b_4 + 6b_1b_2b_3b_4b_5)
p_{\texttt{ACGT}} &= \frac{1}{2}(a_1a_3b_2b_4b_5 + a_1a_4b_2b_3b_5 + a_1a_5b_2b_3b_4 + a_1b_2b_3b_4b_5 + a_2a_3b_1b_4b_5 + a_2a_4b_1b_3b_5 + a_2a_5b_1b_3b_4 + a_2b_1b_3b_4b_5 + a_3a_5b_1b_2b_4 + a_3b_1b_2b_4b_5 + a_4a_5b_1b_2b_3 + a_4b_1b_2b_3b_5 + 4b_1b_2b_3b_4b_5)
p_{\texttt{ACCA}} &= \frac{1}{2}(a_1a_3b_2b_4b_5 + a_1a_4a_5b_2b_3 + 2a_1b_2b_3b_4b_5 + a_2a_3a_5b_1b_4 + a_2a_4b_1b_3b_5 + 2a_2b_1b_3b_4b_5 + 2a_3b_1b_2b_4b_5 + 2a_4b_1b_2b_3b_5 + 2a_5b_1b_2b_3b_4 + 2b_1b_2b_3b_4b_5)
\end{align}$
Fourier parametrization
$\begin{align}
q_{\texttt{CGAT}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{2} \\[0.5em]
q_{\texttt{CCCC}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{1} \\[0.5em]
q_{\texttt{ACAC}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{2} \\[0.5em]
q_{\texttt{CGTA}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{2} \\[0.5em]
q_{\texttt{CCAA}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{1}x^{(4)}_{1}x^{(5)}_{1} \\[0.5em]
q_{\texttt{AACC}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{1} \\[0.5em]
q_{\texttt{CGCG}} &= x^{(1)}_{2}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{2} \\[0.5em]
q_{\texttt{CACA}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{2} \\[0.5em]
q_{\texttt{CAGT}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{2} \\[0.5em]
q_{\texttt{AAAA}} &= x^{(1)}_{1}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{1}x^{(5)}_{1} \\[0.5em]
q_{\texttt{ACGT}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{2}x^{(5)}_{2} \\[0.5em]
q_{\texttt{CAAC}} &= x^{(1)}_{2}x^{(2)}_{1}x^{(3)}_{1}x^{(4)}_{2}x^{(5)}_{2} \\[0.5em]
q_{\texttt{ACCA}} &= x^{(1)}_{1}x^{(2)}_{2}x^{(3)}_{2}x^{(4)}_{1}x^{(5)}_{2}
\end{align}$
Equivalent classes of probability parametrization
$\begin{align}
\text{Class } &p_{\texttt{AAAA}}:\ p_{\texttt{AAAA}},\ p_{\texttt{CCCC}},\ p_{\texttt{GGGG}},\ p_{\texttt{TTTT}} \\[0.5em]
\text{Class } &p_{\texttt{AAAC}}:\ p_{\texttt{AAAC}},\ p_{\texttt{AAAG}},\ p_{\texttt{AAAT}},\ p_{\texttt{CCCA}},\ p_{\texttt{CCCG}},\ p_{\texttt{CCCT}},\ p_{\texttt{GGGA}},\ p_{\texttt{GGGC}},\ p_{\texttt{GGGT}},\ p_{\texttt{TTTA}},\ p_{\texttt{TTTC}},\ p_{\texttt{TTTG}} \\[0.5em]
\text{Class } &p_{\texttt{AACA}}:\ p_{\texttt{AACA}},\ p_{\texttt{AAGA}},\ p_{\texttt{AATA}},\ p_{\texttt{CCAC}},\ p_{\texttt{CCGC}},\ p_{\texttt{CCTC}},\ p_{\texttt{GGAG}},\ p_{\texttt{GGCG}},\ p_{\texttt{GGTG}},\ p_{\texttt{TTAT}},\ p_{\texttt{TTCT}},\ p_{\texttt{TTGT}} \\[0.5em]
\text{Class } &p_{\texttt{AACC}}:\ p_{\texttt{AACC}},\ p_{\texttt{AAGG}},\ p_{\texttt{AATT}},\ p_{\texttt{CCAA}},\ p_{\texttt{CCGG}},\ p_{\texttt{CCTT}},\ p_{\texttt{GGAA}},\ p_{\texttt{GGCC}},\ p_{\texttt{GGTT}},\ p_{\texttt{TTAA}},\ p_{\texttt{TTCC}},\ p_{\texttt{TTGG}} \\[0.5em]
\text{Class } &p_{\texttt{AACG}}:\ p_{\texttt{AACG}},\ p_{\texttt{AACT}},\ p_{\texttt{AAGC}},\ p_{\texttt{AAGT}},\ p_{\texttt{AATC}},\ p_{\texttt{AATG}},\ p_{\texttt{CCAG}},\ p_{\texttt{CCAT}},\ p_{\texttt{CCGA}},\ p_{\texttt{CCGT}},\ p_{\texttt{CCTA}},\ p_{\texttt{CCTG}}\\[0.5em]
\text{Class } &p_{\texttt{ACAA}}:\ p_{\texttt{ACAA}},\ p_{\texttt{AGAA}},\ p_{\texttt{ATAA}},\ p_{\texttt{CACC}},\ p_{\texttt{CGCC}},\ p_{\texttt{CTCC}},\ p_{\texttt{GAGG}},\ p_{\texttt{GCGG}},\ p_{\texttt{GTGG}},\ p_{\texttt{TATT}},\ p_{\texttt{TCTT}},\ p_{\texttt{TGTT}} \\[0.5em]
\text{Class } &p_{\texttt{ACAC}}:\ p_{\texttt{ACAC}},\ p_{\texttt{AGAG}},\ p_{\texttt{ATAT}},\ p_{\texttt{CACA}},\ p_{\texttt{CGCG}},\ p_{\texttt{CTCT}},\ p_{\texttt{GAGA}},\ p_{\texttt{GCGC}},\ p_{\texttt{GTGT}},\ p_{\texttt{TATA}},\ p_{\texttt{TCTC}},\ p_{\texttt{TGTG}} \\[0.5em]
\text{Class } &p_{\texttt{ACAG}}:\ p_{\texttt{ACAG}},\ p_{\texttt{ACAT}},\ p_{\texttt{AGAC}},\ p_{\texttt{AGAT}},\ p_{\texttt{ATAC}},\ p_{\texttt{ATAG}},\ p_{\texttt{CACG}},\ p_{\texttt{CACT}},\ p_{\texttt{CGCA}},\ p_{\texttt{CGCT}},\ p_{\texttt{CTCA}},\ p_{\texttt{CTCG}}, \\[0.5em]
\text{Class } &p_{\texttt{ACCA}}:\ p_{\texttt{ACCA}},\ p_{\texttt{AGGA}},\ p_{\texttt{ATTA}},\ p_{\texttt{CAAC}},\ p_{\texttt{CGGC}},\ p_{\texttt{CTTC}},\ p_{\texttt{GAAG}},\ p_{\texttt{GCCG}},\ p_{\texttt{GTTG}},\ p_{\texttt{TAAT}},\ p_{\texttt{TCCT}},\ p_{\texttt{TGGT}} \\[0.5em]
\text{Class } &p_{\texttt{ACCC}}:\ p_{\texttt{ACCC}},\ p_{\texttt{AGGG}},\ p_{\texttt{ATTT}},\ p_{\texttt{CAAA}},\ p_{\texttt{CGGG}},\ p_{\texttt{CTTT}},\ p_{\texttt{GAAA}},\ p_{\texttt{GCCC}},\ p_{\texttt{GTTT}},\ p_{\texttt{TAAA}},\ p_{\texttt{TCCC}},\ p_{\texttt{TGGG}} \\[0.5em]
\text{Class } &p_{\texttt{ACCG}}:\ p_{\texttt{ACCG}},\ p_{\texttt{ACCT}},\ p_{\texttt{AGGC}},\ p_{\texttt{AGGT}},\ p_{\texttt{ATTC}},\ p_{\texttt{ATTG}},\ p_{\texttt{CAAG}},\ p_{\texttt{CAAT}},\ p_{\texttt{CGGA}},\ p_{\texttt{CGGT}},\ p_{\texttt{CTTA}},\ p_{\texttt{CTTG}}, \\[0.5em]
\text{Class } &p_{\texttt{ACGA}}:\ p_{\texttt{ACGA}},\ p_{\texttt{ACTA}},\ p_{\texttt{AGCA}},\ p_{\texttt{AGTA}},\ p_{\texttt{ATCA}},\ p_{\texttt{ATGA}},\ p_{\texttt{CAGC}},\ p_{\texttt{CATC}},\ p_{\texttt{CGAC}},\ p_{\texttt{CGTC}},\ p_{\texttt{CTAC}},\ p_{\texttt{CTGC}}, \\[0.5em]
\text{Class } &p_{\texttt{ACGC}}:\ p_{\texttt{ACGC}},\ p_{\texttt{ACTC}},\ p_{\texttt{AGCG}},\ p_{\texttt{AGTG}},\ p_{\texttt{ATCT}},\ p_{\texttt{ATGT}},\ p_{\texttt{CAGA}},\ p_{\texttt{CATA}},\ p_{\texttt{CGAG}},\ p_{\texttt{CGTG}},\ p_{\texttt{CTAT}},\ p_{\texttt{CTGT}}, \\[0.5em]
\text{Class } &p_{\texttt{ACGG}}:\ p_{\texttt{ACGG}},\ p_{\texttt{ACTT}},\ p_{\texttt{AGCC}},\ p_{\texttt{AGTT}},\ p_{\texttt{ATCC}},\ p_{\texttt{ATGG}},\ p_{\texttt{CAGG}},\ p_{\texttt{CATT}},\ p_{\texttt{CGAA}},\ p_{\texttt{CGTT}},\ p_{\texttt{CTAA}},\ p_{\texttt{CTGG}}, \\[0.5em]
\text{Class } &p_{\texttt{ACGT}}:\ p_{\texttt{ACGT}},\ p_{\texttt{ACTG}},\ p_{\texttt{AGCT}},\ p_{\texttt{AGTC}},\ p_{\texttt{ATCG}},\ p_{\texttt{ATGC}},\ p_{\texttt{CAGT}},\ p_{\texttt{CATG}},\ p_{\texttt{CGAT}},\ p_{\texttt{CGTA}},\ p_{\texttt{CTAG}},\ p_{\texttt{CTGA}}
\end{align}$
Equivalent classes of Fourier parametrization
$\begin{align}
\text{Class } &0:\ q_{\texttt{CAAA}}, q_{\texttt{GAAA}}, q_{\texttt{TAAA}}, q_{\texttt{ACAA}}, q_{\texttt{GCAA}}, q_{\texttt{TCAA}}, q_{\texttt{AGAA}}, q_{\texttt{CGAA}}, q_{\texttt{TGAA}}, q_{\texttt{ATAA}}, q_{\texttt{CTAA}}, q_{\texttt{GTAA}}, q_{\texttt{AACA}}, q_{\texttt{GACA}}, q_{\texttt{TACA}}, q_{\texttt{CCCA}}, q_{\texttt{GCCA}}, q_{\texttt{TCCA}}, q_{\texttt{AGCA}}, q_{\texttt{CGCA}}, q_{\texttt{GGCA}}, q_{\texttt{ATCA}}, q_{\texttt{CTCA}}, q_{\texttt{TTCA}}, q_{\texttt{AAGA}}, q_{\texttt{CAGA}}, q_{\texttt{TAGA}}, q_{\texttt{ACGA}}, q_{\texttt{CCGA}}, q_{\texttt{GCGA}}, q_{\texttt{CGGA}}, q_{\texttt{GGGA}}, q_{\texttt{TGGA}}, q_{\texttt{ATGA}}, q_{\texttt{GTGA}}, q_{\texttt{TTGA}}, q_{\texttt{AATA}}, q_{\texttt{CATA}}, q_{\texttt{GATA}}, q_{\texttt{ACTA}} \\[0.5em]
\text{Class } &q_{\texttt{AAAA}}:\ q_{\texttt{AAAA}} \\[0.5em]
\text{Class } &q_{\texttt{AACC}}:\ q_{\texttt{AACC}},\ q_{\texttt{AAGG}},\ q_{\texttt{AATT}} \\[0.5em]
\text{Class } &q_{\texttt{ACAC}}:\ q_{\texttt{ACAC}},\ q_{\texttt{AGAG}},\ q_{\texttt{ATAT}} \\[0.5em]
\text{Class } &q_{\texttt{ACCA}}:\ q_{\texttt{ACCA}},\ q_{\texttt{AGGA}},\ q_{\texttt{ATTA}} \\[0.5em]
\text{Class } &q_{\texttt{ACGT}}:\ q_{\texttt{ACGT}},\ q_{\texttt{ACTG}},\ q_{\texttt{AGCT}},\ q_{\texttt{AGTC}},\ q_{\texttt{ATCG}},\ q_{\texttt{ATGC}} \\[0.5em]
\text{Class } &q_{\texttt{CAAC}}:\ q_{\texttt{CAAC}},\ q_{\texttt{GAAG}},\ q_{\texttt{TAAT}} \\[0.5em]
\text{Class } &q_{\texttt{CACA}}:\ q_{\texttt{CACA}},\ q_{\texttt{GAGA}},\ q_{\texttt{TATA}} \\[0.5em]
\text{Class } &q_{\texttt{CAGT}}:\ q_{\texttt{CAGT}},\ q_{\texttt{CATG}},\ q_{\texttt{GACT}},\ q_{\texttt{GATC}},\ q_{\texttt{TACG}},\ q_{\texttt{TAGC}} \\[0.5em]
\text{Class } &q_{\texttt{CCAA}}:\ q_{\texttt{CCAA}},\ q_{\texttt{GGAA}},\ q_{\texttt{TTAA}} \\[0.5em]
\text{Class } &q_{\texttt{CCCC}}:\ q_{\texttt{CCCC}},\ q_{\texttt{CCGG}},\ q_{\texttt{CCTT}},\ q_{\texttt{GGCC}},\ q_{\texttt{GGGG}},\ q_{\texttt{GGTT}},\ q_{\texttt{TTCC}},\ q_{\texttt{TTGG}},\ q_{\texttt{TTTT}} \\[0.5em]
\text{Class } &q_{\texttt{CGAT}}:\ q_{\texttt{CGAT}},\ q_{\texttt{CTAG}},\ q_{\texttt{GCAT}},\ q_{\texttt{GTAC}},\ q_{\texttt{TCAG}},\ q_{\texttt{TGAC}} \\[0.5em]
\text{Class } &q_{\texttt{CGCG}}:\ q_{\texttt{CGCG}},\ q_{\texttt{CGGC}},\ q_{\texttt{CTCT}},\ q_{\texttt{CTTC}},\ q_{\texttt{GCCG}},\ q_{\texttt{GCGC}},\ q_{\texttt{GTGT}},\ q_{\texttt{GTTG}},\ q_{\texttt{TCCT}},\ q_{\texttt{TCTC}},\ q_{\texttt{TGGT}},\ q_{\texttt{TGTG}} \\[0.5em]
\text{Class } &q_{\texttt{CGTA}}:\ q_{\texttt{CGTA}},\ q_{\texttt{CTGA}},\ q_{\texttt{GCTA}},\ q_{\texttt{GTCA}},\ q_{\texttt{TCGA}},\ q_{\texttt{TGCA}}
\end{align}$
Invariants in Fourier coordinates
Ideal generated by $$ \begin{align}
q_{CACA}q_{ACGT} - q_{CAGT}q_{ACCA} \\
q_{CGTA}q_{ACGT} - q_{CGCG}q_{ACCA} \\
q_{ACAC}q_{CAGT} - q_{ACGT}q_{CAAC} \\
q_{CGAT}q_{CAGT} - q_{CGCG}q_{CAAC} \\
-q_{CGTA}q_{CAGT} + q_{CGCG}q_{CACA} \\
q_{ACAC}q_{CACA} - q_{CAAC}q_{ACCA} \\
q_{CGAT}q_{CACA} - q_{CGTA}q_{CAAC} \\
-q_{CGAT}q_{ACGT} + q_{ACAC}q_{CGCG} \\
-q_{CCCC}q_{AAAA} + q_{CCAA}q_{AACC} \\
-q_{CGAT}q_{ACCA} + q_{ACAC}q_{CGTA} \\
-q_{ACAC}q_{AACC}q_{ACCA} + q_{AAAA}q_{ACGT}^2 \\
-q_{CCCC}q_{ACAC}q_{ACCA} + q_{CCAA}q_{ACGT}^2 \\
-q_{AACC}q_{CAAC}q_{ACCA} + q_{CAGT}q_{AAAA}q_{ACGT} \\
-q_{CGAT}q_{AACC}q_{ACCA} + q_{CGCG}q_{AAAA}q_{ACGT} \\
-q_{CCCC}q_{CAAC}q_{ACCA} + q_{CCAA}q_{CAGT}q_{ACGT} \\
-q_{CGAT}q_{CCCC}q_{ACCA} + q_{CCAA}q_{CGCG}q_{ACGT} \\
-q_{AACC}q_{CACA}q_{CAAC} + q_{CAGT}^2q_{AAAA} \\
-q_{CGTA}q_{AACC}q_{CAAC} + q_{CGCG}q_{CAGT}q_{AAAA} \\
-q_{CCCC}q_{CAAC}q_{ACCA} + q_{CGCG}^2q_{AAAA} \\
q_{CGTA}q_{CGCG}q_{AAAA} - q_{CCAA}q_{CAGT}q_{ACCA} \\
q_{CGAT}q_{CGCG}q_{AAAA} - q_{CCAA}q_{ACGT}q_{CAAC} \\
q_{CGTA}^2q_{AAAA} - q_{CCAA}q_{CACA}q_{ACCA} \\
q_{CGAT}q_{CGTA}q_{AAAA} - q_{CCAA}q_{CAAC}q_{ACCA} \\
q_{CGAT}^2q_{AAAA} - q_{ACAC}q_{CCAA}q_{CAAC} \\
-q_{CCCC}q_{CACA}q_{CAAC} + q_{CCAA}q_{CAGT}^2 \\
-q_{CCCC}q_{CGTA}q_{CAAC} + q_{CCAA}q_{CGCG}q_{CAGT} \\
-q_{CCCC}q_{CAGT}q_{ACGT} + q_{AACC}q_{CGCG}^2 \\
-q_{CCCC}q_{CAGT}q_{ACCA} + q_{CGTA}q_{AACC}q_{CGCG} \\
q_{CGAT}q_{AACC}q_{CGCG} - q_{CCCC}q_{ACGT}q_{CAAC} \\
-q_{CCCC}q_{CACA}q_{ACCA} + q_{CGTA}^2q_{AACC} \\
q_{CGAT}q_{CGTA}q_{AACC} - q_{CCCC}q_{CAAC}q_{ACCA} \\
q_{CGAT}^2q_{AACC} - q_{CCCC}q_{ACAC}q_{CAAC} \\
q_{CGAT}q_{CCCC}q_{CGTA} - q_{CCAA}q_{CGCG}^2
\end{align} $$
Gröbner basis of ideal of invariants
Gröbner basis with elements $$ \begin{align}
q_{CACA}q_{ACGT} - q_{CAGT}q_{ACCA} \\
q_{CGTA}q_{ACGT} - q_{CGCG}q_{ACCA} \\
q_{ACAC}q_{CAGT} - q_{ACGT}q_{CAAC} \\
q_{CGAT}q_{CAGT} - q_{CGCG}q_{CAAC} \\
-q_{CGTA}q_{CAGT} + q_{CGCG}q_{CACA} \\
q_{ACAC}q_{CACA} - q_{CAAC}q_{ACCA} \\
q_{CGAT}q_{CACA} - q_{CGTA}q_{CAAC} \\
-q_{CGAT}q_{ACGT} + q_{ACAC}q_{CGCG} \\
-q_{CCCC}q_{AAAA} + q_{CCAA}q_{AACC} \\
-q_{CGAT}q_{ACCA} + q_{ACAC}q_{CGTA} \\
-q_{ACAC}q_{AACC}q_{ACCA} + q_{AAAA}q_{ACGT}^2 \\
-q_{CCCC}q_{ACAC}q_{ACCA} + q_{CCAA}q_{ACGT}^2 \\
-q_{AACC}q_{CAAC}q_{ACCA} + q_{CAGT}q_{AAAA}q_{ACGT} \\
-q_{CGAT}q_{AACC}q_{ACCA} + q_{CGCG}q_{AAAA}q_{ACGT} \\
-q_{CCCC}q_{CAAC}q_{ACCA} + q_{CCAA}q_{CAGT}q_{ACGT} \\
-q_{CGAT}q_{CCCC}q_{ACCA} + q_{CCAA}q_{CGCG}q_{ACGT} \\
-q_{AACC}q_{CACA}q_{CAAC} + q_{CAGT}^2q_{AAAA} \\
-q_{CGTA}q_{AACC}q_{CAAC} + q_{CGCG}q_{CAGT}q_{AAAA} \\
-q_{CCCC}q_{CAAC}q_{ACCA} + q_{CGCG}^2q_{AAAA} \\
q_{CGTA}q_{CGCG}q_{AAAA} - q_{CCAA}q_{CAGT}q_{ACCA} \\
q_{CGAT}q_{CGCG}q_{AAAA} - q_{CCAA}q_{ACGT}q_{CAAC} \\
q_{CGTA}^2q_{AAAA} - q_{CCAA}q_{CACA}q_{ACCA} \\
q_{CGAT}q_{CGTA}q_{AAAA} - q_{CCAA}q_{CAAC}q_{ACCA} \\
q_{CGAT}^2q_{AAAA} - q_{ACAC}q_{CCAA}q_{CAAC} \\
-q_{CCCC}q_{CACA}q_{CAAC} + q_{CCAA}q_{CAGT}^2 \\
-q_{CCCC}q_{CGTA}q_{CAAC} + q_{CCAA}q_{CGCG}q_{CAGT} \\
-q_{CCCC}q_{CAGT}q_{ACGT} + q_{AACC}q_{CGCG}^2 \\
-q_{CCCC}q_{CAGT}q_{ACCA} + q_{CGTA}q_{AACC}q_{CGCG} \\
q_{CGAT}q_{AACC}q_{CGCG} - q_{CCCC}q_{ACGT}q_{CAAC} \\
-q_{CCCC}q_{CACA}q_{ACCA} + q_{CGTA}^2q_{AACC} \\
q_{CGAT}q_{CGTA}q_{AACC} - q_{CCCC}q_{CAAC}q_{ACCA} \\
q_{CGAT}^2q_{AACC} - q_{CCCC}q_{ACAC}q_{CAAC} \\
q_{CGAT}q_{CCCC}q_{CGTA} - q_{CCAA}q_{CGCG}^2
\end{align}$$
with respect to the ordering
degrevlex $ q_{\texttt{CGAT}}, q_{\texttt{CCCC}}, q_{\texttt{ACAC}}, q_{\texttt{CGTA}}, q_{\texttt{CCAA}}, q_{\texttt{AACC}}, q_{\texttt{CGCG}}, q_{\texttt{CACA}}, q_{\texttt{CAGT}}, q_{\texttt{AAAA}}, q_{\texttt{ACGT}}, q_{\texttt{CAAC}}, q_{\texttt{ACCA}}$
Additional information
The cardinality of the smallest set of generators that define the ideal of phylogenetic invariants: 33
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: 3
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: 3
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: 33
The largest degree of a generator in the degree reverse lexicographic reduced Gröbner basis: 3
The largest degree of an element in a minimal generating set for the ideal of phylogenetic invariants: 3
The cardinality of the degree reverse lexicographic reduced Gröbner basis of the ideal of phylogenetic invariants: 33
Specialized Fourier transform
$\frac{1}{9} \begin{pmatrix}
9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9\\
9 & -3 & -3 & 9 & -3 & 9 & -3 & -3 & -3 & 9 & -3 & -3 & -3 & 9 & -3\\
9 & -3 & 9 & -3 & -3 & -3 & 9 & -3 & -3 & 9 & -3 & -3 & 9 & -3 & -3\\
9 & 9 & -3 & -3 & -3 & -3 & -3 & -3 & 9 & 9 & 9 & -3 & -3 & -3 & -3\\
9 & -3 & -3 & -3 & 3 & -3 & -3 & 3 & -3 & 9 & -3 & 3 & -3 & -3 & 3\\
9 & -3 & 9 & -3 & -3 & 9 & -3 & -3 & 9 & -3 & -3 & 9 & -3 & -3 & -3\\
9 & 9 & -3 & -3 & -3 & 9 & 9 & 9 & -3 & -3 & -3 & -3 & -3 & -3 & -3\\
9 & -3 & -3 & -3 & 3 & 9 & -3 & -3 & -3 & -3 & 3 & -3 & 3 & -3 & 3\\
9 & 9 & 9 & 9 & 9 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & -3\\
9 & -3 & -3 & 9 & -3 & -3 & 1 & 1 & 1 & -3 & 1 & 1 & 1 & -3 & 1\\
9 & -3 & 9 & -3 & -3 & -3 & -3 & 3 & -3 & -3 & 3 & -3 & -3 & 3 & 3\\
9 & -3 & -3 & -3 & 3 & -3 & 3 & 0 & 3 & -3 & 0 & 0 & 0 & 3 & -3\\
9 & 9 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & -3 & 3 & 3 & 3 & 3
\end{pmatrix} ] $
Inverse specialized Fourier transform
$\frac{1}{64} \begin{pmatrix}
1 & 3 & 3 & 3 & 6 & 3 & 3 & 6 & 3 & 9 & 6 & 12 & 6\\
3 & -3 & -3 & 9 & -6 & -3 & 9 & -6 & 9 & -9 & -6 & -12 & 18\\
3 & -3 & 9 & -3 & -6 & 9 & -3 & -6 & 9 & -9 & 18 & -12 & -6\\
3 & 9 & -3 & -3 & -6 & -3 & -3 & -6 & 9 & 27 & -6 & -12 & -6\\
6 & -6 & -6 & -6 & 12 & -6 & -6 & 12 & 18 & -18 & -12 & 24 & -12\\
3 & 9 & -3 & -3 & -6 & 9 & 9 & 18 & -3 & -9 & -6 & -12 & -6\\
3 & -3 & 9 & -3 & -6 & -3 & 9 & -6 & -3 & 3 & -6 & 12 & -6\\
6 & -6 & -6 & -6 & 12 & -6 & 18 & -12 & -6 & 6 & 12 & 0 & -12\\
3 & -3 & -3 & 9 & -6 & 9 & -3 & -6 & -3 & 3 & -6 & 12 & -6\\
3 & 9 & 9 & 9 & 18 & -3 & -3 & -6 & -3 & -9 & -6 & -12 & -6\\
6 & -6 & -6 & 18 & -12 & -6 & -6 & 12 & -6 & 6 & 12 & 0 & -12\\
6 & -6 & -6 & -6 & 12 & 18 & -6 & -12 & -6 & 6 & -12 & 0 & 12\\
6 & -6 & 18 & -6 & -12 & -6 & -6 & 12 & -6 & 6 & -12 & 0 & 12\\
6 & 18 & -6 & -6 & -12 & -6 & -6 & -12 & -6 & -18 & 12 & 24 & 12\\
6 & -6 & -6 & -6 & 12 & -6 & -6 & 12 & -6 & 6 & 12 & -24 & 12
\end{pmatrix} $