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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 72450.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.ca1 | 72450q2 | \([1, -1, 0, -125727, -16950619]\) | \(89332607016927/1060723384\) | \(2609777295909000\) | \([2]\) | \(442368\) | \(1.7698\) | |
| 72450.ca2 | 72450q1 | \([1, -1, 0, -1527, -680419]\) | \(-160103007/81288256\) | \(-199999592856000\) | \([2]\) | \(221184\) | \(1.4232\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.ca do not have complex multiplication.Modular form 72450.2.a.ca
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
