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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 8712.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8712.a1 | 8712n1 | \([0, 0, 0, -8729787, -9927809210]\) | \(55635379958596/24057\) | \(31814497208558592\) | \([2]\) | \(322560\) | \(2.5090\) | \(\Gamma_0(N)\)-optimal |
| 8712.a2 | 8712n2 | \([0, 0, 0, -8686227, -10031786930]\) | \(-27403349188178/578739249\) | \(-1530722718692588095488\) | \([2]\) | \(645120\) | \(2.8555\) |
Rank
sage: E.rank()
The elliptic curves in class 8712.a have rank \(1\).
Complex multiplication
The elliptic curves in class 8712.a do not have complex multiplication.Modular form 8712.2.a.a
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.
