# Properties

 Label 112710cj Number of curves 8 Conductor 112710 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("112710.cj1")

sage: E.isogeny_class()

## Elliptic curves in class 112710cj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
112710.cj7 112710cj1 [1, 1, 1, -605786200, 5525820002585] [4] 79626240 $$\Gamma_0(N)$$-optimal
112710.cj6 112710cj2 [1, 1, 1, -1606049880, -17425830293223] [2, 2] 159252480
112710.cj5 112710cj3 [1, 1, 1, -7454115160, -246099036985063] [4] 238878720
112710.cj8 112710cj4 [1, 1, 1, 4300555240, -115815695740135] [2] 318504960
112710.cj4 112710cj5 [1, 1, 1, -23516873880, -1387921578504423] [2] 318504960
112710.cj2 112710cj6 [1, 1, 1, -119049939240, -15810413259752295] [2, 2] 477757440
112710.cj3 112710cj7 [1, 1, 1, -118834062020, -15870608118373543] [2] 955514880
112710.cj1 112710cj8 [1, 1, 1, -1904799001740, -1011864096741752295] [2] 955514880

## Rank

sage: E.rank()

The elliptic curves in class 112710cj have rank $$0$$.

## Modular form 112710.2.a.cj

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + q^{13} + 4q^{14} - q^{15} + q^{16} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.