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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 42588e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42588.l4 | 42588e1 | \([0, 0, 0, 10140, 134017]\) | \(2048000/1323\) | \(-74484767932848\) | \([2]\) | \(110592\) | \(1.3507\) | \(\Gamma_0(N)\)-optimal |
| 42588.l3 | 42588e2 | \([0, 0, 0, -43095, 1102894]\) | \(9826000/5103\) | \(4596774249570048\) | \([2]\) | \(221184\) | \(1.6973\) | |
| 42588.l2 | 42588e3 | \([0, 0, 0, -172380, 28369861]\) | \(-10061824000/352947\) | \(-19870880867418672\) | \([2]\) | \(331776\) | \(1.9000\) | |
| 42588.l1 | 42588e4 | \([0, 0, 0, -2780895, 1784943862]\) | \(2640279346000/3087\) | \(2780764669492992\) | \([2]\) | \(663552\) | \(2.2466\) |
Rank
sage: E.rank()
The elliptic curves in class 42588e have rank \(1\).
Complex multiplication
The elliptic curves in class 42588e do not have complex multiplication.Modular form 42588.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
