AIMS-7 Galois Groups

1,686,353

Total Polynomials

AIMS-7 dataset

7

Galois Groups

transitive subgroups of S7

92.5%

Are S7

1,559,957 entries

0.8525

ML Balanced Acc.

non-S7 + invariants model

252

C7 Polynomials

rarest group in AIMS-7

h = 28

First C7 Example

from p=29 cyclotomic field

AIMS-7 Full Dataset -- Real Counts from AIMS-7.csv (Table 9)

Full Dataset Distribution (including S7)

Non-S7 Distribution (126,396 polynomials)

Complete Reference Table -- AIMS-7.csv + AIMS-7inv.csv

Group	Order	AIMS-7 Count	LMFDB Count	h le 4	R1 factors	R2 factors	R3 factors	Solvable

Subgroup Inclusion Lattice -- click any node

◈Click any node to explore that Galois group — order, resolvent factorizations, and real AIMS-7 counts.

Mean Log-Height E[h(f)|G] -- Figure 5 of paper

Counts at Height le 4 -- Table 6 (log scale)

Rational vs Irreducible Points by Height -- Table 4

LMFDB vs AIMS-7 Final Counts (non-S7 groups)

Key Statistical Findings

Height-Complexity Trend

Counter-intuitively, higher group complexity is associated with smaller defining height. A7 and L(3,2) live at small heights; C7 and C7xC3 appear only at large heights.

Discriminant Correlation

Large heights accompany large |Delta_f|. The C7 and C7xC3 points in the joint (h, log|Delta|) plot cluster in the upper-right, far from A7 and L(3,2).

Cyclic Group Rarity

C7 and C7xC3 are completely absent at height le 4. The first C7 polynomial appears at h=28, from the cyclotomic field Q(zeta_29), derived from p=29.

Classification Pipeline -- AIMS-7.csv + AIMS-7inv.2025.csv

AIMS-7.csv
1.69M entries

+

AIMS-7inv.csv
j0-j4, Delta_f

→

Coefficients
a0...a7

→

Hist. Gradient
Boosting
60/40 split

→

Galois Group
Label
7 classes

Model Comparison -- Balanced Accuracy

Non-S7 Model: Precision / Recall / F1 (Table 10)

Confusion Matrix -- Non-S7 Test Set, 50,559 polynomials (Table 11)

Rows = True label · Columns = Predicted · Green diagonal = correct · Red = major errors

Main confusion: 2,616 true A7 classified as L(3,2), and 5,923 true L(3,2) classified as A7. All other group pairs are well-separated.

Feature Importance (Neurosymbolic Model)

5-Fold Cross-Validation Balanced Accuracy

Neurosymbolic Network Architecture (Section 8)

1 Symbolic
Preprocess
Delta, j0-j4
2 Feature
Transform
sgn*log10
3 Concat
15-vector
coeff+inv
4 Dense
128->64
ReLU
5 Self-
Attention
dynamic
6 Softmax
Output
6 non-S7