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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 57960bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57960.bo3 | 57960bc1 | \([0, 0, 0, -845409207, 9461251202506]\) | \(358061097267989271289240144/176126855625\) | \(32869498304160000\) | \([2]\) | \(8847360\) | \(3.4065\) | \(\Gamma_0(N)\)-optimal |
| 57960.bo2 | 57960bc2 | \([0, 0, 0, -845413707, 9461145444406]\) | \(89516703758060574923008036/1985322833430374025\) | \(1482035553864440488166400\) | \([2, 2]\) | \(17694720\) | \(3.7531\) | |
| 57960.bo4 | 57960bc3 | \([0, 0, 0, -815233107, 10167908699086]\) | \(-40133926989810174413190818/6689384645060302103835\) | \(-9987197759997870558608824320\) | \([2]\) | \(35389440\) | \(4.0996\) | |
| 57960.bo1 | 57960bc4 | \([0, 0, 0, -875666307, 8747613671326]\) | \(49737293673675178002921218/6641736806881023047235\) | \(9916059918778912361337477120\) | \([2]\) | \(35389440\) | \(4.0996\) |
Rank
sage: E.rank()
The elliptic curves in class 57960bc have rank \(1\).
Complex multiplication
The elliptic curves in class 57960bc do not have complex multiplication.Modular form 57960.2.a.bc
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
