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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 390390ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.ei4 | 390390ei1 | \([1, 0, 0, -160976, 33265920]\) | \(-95575628340361/43812679680\) | \(-211475436593541120\) | \([2]\) | \(7077888\) | \(2.0309\) | \(\Gamma_0(N)\)-optimal |
| 390390.ei3 | 390390ei2 | \([1, 0, 0, -2810896, 1813482176]\) | \(508859562767519881/62240270400\) | \(300421897329153600\) | \([2, 2]\) | \(14155776\) | \(2.3774\) | |
| 390390.ei1 | 390390ei3 | \([1, 0, 0, -44973016, 116081259800]\) | \(2084105208962185000201/31185000\) | \(150524038665000\) | \([2]\) | \(28311552\) | \(2.7240\) | |
| 390390.ei2 | 390390ei4 | \([1, 0, 0, -3047496, 1490144616]\) | \(648474704552553481/176469171805080\) | \(851782986691306389720\) | \([2]\) | \(28311552\) | \(2.7240\) |
Rank
sage: E.rank()
The elliptic curves in class 390390ei have rank \(1\).
Complex multiplication
The elliptic curves in class 390390ei do not have complex multiplication.Modular form 390390.2.a.ei
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.
