# Properties

 Label 57330ce Number of curves 8 Conductor 57330 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("57330.cg1")

sage: E.isogeny_class()

## Elliptic curves in class 57330ce

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57330.cg7 57330ce1 [1, -1, 0, -11344734, 14631518740] [2] 3538944 $$\Gamma_0(N)$$-optimal
57330.cg6 57330ce2 [1, -1, 0, -18259614, -5306846252] [2, 2] 7077888
57330.cg5 57330ce3 [1, -1, 0, -70085934, -215981034620] [2] 10616832
57330.cg8 57330ce4 [1, -1, 0, 71757306, -42159773300] [2] 14155776
57330.cg4 57330ce5 [1, -1, 0, -218914614, -1244592257252] [2] 14155776
57330.cg2 57330ce6 [1, -1, 0, -1107750114, -14190619612352] [2, 2] 21233664
57330.cg3 57330ce7 [1, -1, 0, -1094136444, -14556410757050] [2] 42467328
57330.cg1 57330ce8 [1, -1, 0, -17723990664, -908214149412662] [2] 42467328

## Rank

sage: E.rank()

The elliptic curves in class 57330ce have rank $$0$$.

## Modular form 57330.2.a.cg

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - q^{13} + q^{16} + 6q^{17} + 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.