Consider the following polynomial $p$ in three variables, a main one called $x$ and two secondary ones $b$ and $c$. The $x^ib^jc^k$ coefficient is p[i][j][k], using zero-indexing; $p$ is symmetric in $b$ and $c$.
p = \ [[[ 72, 240, 216, -96, -264, -144, -24], [ 240, 368, 96, -288, -336, -80], [ 216, 96, 1104, 480, 24], [ -96, -288, 480, 160], [-264, -336, 24], [-144, -80], [ -24]], [[ -300, -1208, -1872, -992, 888, 1776, 1200, 416, 84, 8], [-1208, -4000, -4288, 2368, 4784, 2144, 192, 0, 8], [-1872, -4288, -8112, -6560, -1200, -640, -336, -32], [ -992, 2368, -6560, -4288, 32, 0, -32], [ 888, 4784, -1200, 32, 504, 48], [ 1776, 2144, -640, 0, 48], [ 1200, 192, -336, -32], [ 416, 0, -32], [ 84, 8], [ 8]], [[ 353, 1859, 4248, 6112, 4034, -2978, -7248, -4840, -1387, -153], [ 1859, 12264, 21224, 3568, -18418, -14680, -4768, -896, -153], [ 4248, 21224, 42796, 28996, 6736, 4912, 3756, 612], [ 6112, 3568, 28996, 30384, 4696, 896, 612], [ 4034, -18418, 6736, 4696, -4738, -918], [-2978, -14680, 4912, 896, -918], [-7248, -4768, 3756, 612], [-4840, -896, 612], [-1387, -153], [ -153]], [[ 30, -486, -1480, -7320, -17660, -11188, 11224, 17864, 7886, 1130], [ -486, -14960, -36184, -27376, 17228, 30128, 20872, 9136, 1642], [ -1480, -36184, -110664, -93560, -17560, -13320, -15576, -3496], [ -7320, -27376, -93560, -92384, -25416, -9136, -4520], [-17660, 17228, -17560, -25416, 15380, 5244], [-11188, 30128, -13320, -9136, 5244], [ 11224, 20872, -15576, -4520], [ 17864, 9136, -3496], [ 7886, 1642], [ 1130]], [[ -175, -561, -4164, -5996, 15846, 35946, 7612, -25548, -19119, -3841], [ -561, 7800, 23316, 24888, 3498, -2520, -20556, -27864, -8001], [ -4164, 23316, 126036, 157596, 42660, 24012, 35148, 10116], [ -5996, 24888, 157596, 184176, 41292, 27864, 18436], [ 15846, 3498, 42660, 41292, -32058, -16710], [ 35946, -2520, 24012, 27864, -16710], [ 7612, -20556, 35148, 18436], [-25548, -27864, 10116], [-19119, -8001], [ -3841]], [[ -120, -528, 2040, 14232, 8136, -30024, -31128, 10248, 21072, 6072], [ -528, -3504, -9720, 12864, 17688, -40560, -24840, 31200, 17400], [ 2040, -9720, -66960, -101136, -23016, -29592, -59904, -18912], [ 14232, 12864, -101136, -200736, -28776, -31200, -41568], [ 8136, 17688, -23016, -28776, 77664, 37008], [-30024, -40560, -29592, -31200, 37008], [-31128, -24840, -59904, -41568], [ 10248, 31200, -18912], [ 21072, 17400], [ 6072]], [[ 144, 720, 2412, -4572, -15012, 2484, 22212, 5868, -9756, -4500], [ 720, 2304, 12636, -15912, -44748, 24912, 48708, -11304, -17316], [ 2412, 12636, 41256, 12456, -67140, -7668, 69552, 28656], [ -4572, -15912, 12456, 82080, -19836, 11304, 54288], [-15012, -44748, -67140, -19836, -119592, -61128], [ 2484, 24912, -7668, 11304, -61128], [ 22212, 48708, 69552, 54288], [ 5868, -11304, 28656], [ -9756, -17316], [ -4500]], [[ 0, 0, -1296, -1296, 3888, 3888, -3888, -3888, 1296, 1296], [ 0, 0, -6480, 0, 19440, 0, -19440, 0, 6480], [-1296, -6480, -36288, -10368, 81648, 45360, -44064, -28512], [-1296, 0, -10368, 0, 50544, 0, -38880], [ 3888, 19440, 81648, 50544, 85536, 59616], [ 3888, 0, 45360, 0, 59616], [-3888, -19440, -44064, -38880], [-3888, 0, -28512], [ 1296, 6480], [ 1296]], [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 11664, 11664, -23328, -23328, 11664, 11664], [0, 0, 11664, 0, -23328, 0, 11664], [0, 0, -23328, -23328, -23328, -23328], [0, 0, -23328, 0, -23328], [0, 0, 11664, 11664], [0, 0, 11664], [0, 0], [0]]]
This was the first result I obtained in my ultimately successful quest for exact barycentric coordinates of $X(5394)$. Substituting rational numbers for $b$ and $c$ always seemed to yield a polynomial in $\mathbb Q[x]$ with Galois group 8T39 in the general case, so I strongly suspected that the unsubstituted $p$ – as an element of $\mathbb Q(b,c)[x]$ where $\mathbb Q(b,c)$ is the function field of rational functions in $b$ and $c$ – had the same Galois group. If so, there would be a quartic $q\in\mathbb Q(b,c)[x]$ with root $t$ such that $p$ split into a quadratic and sextic over $\mathbb Q(b,c,t)[x]$.
After despairing for a while I realised that a correct $q$ was out there all the time, and through integer relation and interpolating coefficients I obtained the desired quadratic/sextic split; see the linked answer of mine.
Suppose, however, that I had no lead on what $q$ was. Knowing only $p$, how could I use software to
- prove that the Galois group of the unsubstituted $p$ is 8T39, and then
- find a suitable subfield polynomial $q$,
all in reasonable time?
