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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 259210cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259210.cd3 | 259210cd1 | \([1, 1, 1, -3914611, 2892891889]\) | \(380920459249/12622400\) | \(219835180786939726400\) | \([2]\) | \(14598144\) | \(2.6763\) | \(\Gamma_0(N)\)-optimal |
| 259210.cd4 | 259210cd2 | \([1, 1, 1, 1269589, 10003540609]\) | \(12994449551/2489452840\) | \(-43356993530704187539240\) | \([2]\) | \(29196288\) | \(3.0229\) | |
| 259210.cd1 | 259210cd3 | \([1, 1, 1, -43832951, -110752444727]\) | \(534774372149809/5323062500\) | \(92707916642451463062500\) | \([2]\) | \(43794432\) | \(3.2256\) | |
| 259210.cd2 | 259210cd4 | \([1, 1, 1, -11431701, -270801659227]\) | \(-9486391169809/1813439640250\) | \(-31583362210083794630280250\) | \([2]\) | \(87588864\) | \(3.5722\) |
Rank
sage: E.rank()
The elliptic curves in class 259210cd have rank \(0\).
Complex multiplication
The elliptic curves in class 259210cd do not have complex multiplication.Modular form 259210.2.a.cd
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.
