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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 258570.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.e1 | 258570e1 | \([1, -1, 0, -41625, -3075539]\) | \(4980061835533/313344000\) | \(501855823872000\) | \([2]\) | \(1548288\) | \(1.5719\) | \(\Gamma_0(N)\)-optimal |
| 258570.e2 | 258570e2 | \([1, -1, 0, 33255, -12974675]\) | \(2539391358707/46818000000\) | \(-74984317434000000\) | \([2]\) | \(3096576\) | \(1.9184\) |
Rank
sage: E.rank()
The elliptic curves in class 258570.e have rank \(0\).
Complex multiplication
The elliptic curves in class 258570.e do not have complex multiplication.Modular form 258570.2.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 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.
