What we really want from you is a discussion of how you used the AI to get some nice results — the ups and downs and how things played out. Sort of a mingled hacker talk with math research — I hope that makes sense?

$E/\mathbb{Q}$ elliptic curve.

Theorem (Mazur, 1978)

If $E/\mathbb{Q}$ is an elliptic curve admitting a rational $p$-isogeny, then $$p \in \{2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,37,\,43,\,67,\,163\}$$ $$=: \mathrm{IsogPrimeDeg}(\mathbb{Q}).$$

Question (Cremona, 2010)

Mazur did this for $\mathbb{Q}$ in '78 — can you do it for any other number field?

Actually this is a corollary of the following.

Why cubic? Because for every cubic field, $\mathrm{IsogPrimeDeg}(K)$ is finite.

For $p = 29$, there are 5 curves:

$H_1, H_2, H_5$ have genus $3$ (degree $8$); $H_3, H_4$ have genus $2$ (degree $6$).

This paper implemented many techniques for showing that twists of hyperelliptic modular curves don't admit a $\mathbb{Q}$-point (TwoCoverDescent, MWSieve, IsELS, EllipticCurveChabauty, Chabauty0).

We use this to show that many cubic fields $K$ provably do not admit $29$ as an isogeny prime. We add this to a growing ground truth dataset.

Ground-truth dataset

Question: Can we train an ML model to detect cubic fields that admit $29$ as an isogeny prime?

There is a way of generating YESs from plugging in rational $x$-values into one of the $5$ hyperelliptic curves, via a converse to our theorem,

but the discriminants obtained are enormous.

Artificially balancing the dataset with this gave us a dataset of the following distribution:

The features we used in the dataset:

class_group, class_number, conductor, disc_abs, index, monogenic, narrow_class_group, narrow_class_number, num_ram, z1, z2, …, z30

(z1–z30 are the first 30 zeta coefficients.)

We trained (1) XGBoost, (2) Random Forest on this balanced dataset, 80/20 train-test split.

Test set (held-out 20%, 2049 rows, balanced):

Top features (XGBoost):

What did the model actually learn?

index carries 77% of the gain and a 26% PR-AUC drop under permutation — basically the entire signal. Why?

index is almost a perfect class indicator on the balanced training set — but the unknowns share the NO distribution (overwhelmingly $1$).

Predicted on the $1871$ unknowns: only $9$ (XGBoost) / $7$ (RF) flagged YES; median $P(\text{YES}) = 0.0001$.

The model isn't learning the mathematics; it's learning the data provenance — which is a perfect proxy for the label only because we built it that way.

The honest experiment: LMFDB-only

• Train CSV: master_features.csv — 5,314 rows (5,213 NO + 101 YES), ~52:1 imbalance.
• 5-fold stratified CV on labelled rows; metrics on out-of-fold predictions.
• Three models compared: Weighted XGBoost, Focal-loss XGBoost ($\gamma = 2$), Isolation Forest (anomaly detection on NO rows).

Results

Out-of-fold metrics across 5-fold stratified CV (101 YES across all folds):

(Random would put ~$0.95$ YES rows in the top 50.)

• The PR-AUC gap is decisive. ROC-AUC is forgiving under heavy imbalance (it averages over the 5,213 negatives); PR-AUC is the honest metric when YES is rare. Weighted XGBoost beats focal by $0.15$ absolute (~25% relative) on PR-AUC, and beats Isolation Forest by $0.63$ — a different planet.
• Same ordering at every recall@$k$ we measured.
• At top-200 (< 4% of the dataset), Weighted XGBoost captures 76% of all YES rows.
• Isolation Forest treats YES as anomalies — barely beats random, because YES cubics aren't anomalous, just structured.

Top features (Weighted XGBoost)

Top features — permutation importance

disc_abs dominates ($\Delta$ PR-AUC $\approx 0.66$), with z29, class_group_max, z19, z3, regulator following.

⚠ Yellow flag: heavy disc_abs reliance — this drove the rest of the experiments.

Honest vs. cautionary

The first one is the honest result. The second is the cautionary tale.