Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("112710.cj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 112710.cj

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

Rank

sage: E.rank()
 

The elliptic curves in class 112710.cj 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.