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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 134640ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.cp2 | 134640ed1 | \([0, 0, 0, -53403, -4747702]\) | \(152298969481827/86468800\) | \(9562757529600\) | \([2]\) | \(387072\) | \(1.4373\) | \(\Gamma_0(N)\)-optimal |
| 134640.cp3 | 134640ed2 | \([0, 0, 0, -43803, -6508342]\) | \(-84044939142627/116825833960\) | \(-12920002629304320\) | \([2]\) | \(774144\) | \(1.7838\) | |
| 134640.cp1 | 134640ed3 | \([0, 0, 0, -167643, 20834442]\) | \(6462919457883/1414187500\) | \(114014013696000000\) | \([2]\) | \(1161216\) | \(1.9866\) | |
| 134640.cp4 | 134640ed4 | \([0, 0, 0, 372357, 127430442]\) | \(70819203762117/127995282250\) | \(-10319180351597568000\) | \([2]\) | \(2322432\) | \(2.3332\) |
Rank
sage: E.rank()
The elliptic curves in class 134640ed have rank \(0\).
Complex multiplication
The elliptic curves in class 134640ed do not have complex multiplication.Modular form 134640.2.a.ed
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.
