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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1710a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1710.a2 | 1710a1 | \([1, -1, 0, -555, 1925]\) | \(961504803/486400\) | \(9573811200\) | \([2]\) | \(1920\) | \(0.60774\) | \(\Gamma_0(N)\)-optimal |
1710.a1 | 1710a2 | \([1, -1, 0, -4875, -128539]\) | \(651038076963/7220000\) | \(142111260000\) | \([2]\) | \(3840\) | \(0.95431\) |
Rank
sage: E.rank()
The elliptic curves in class 1710a have rank \(1\).
Complex multiplication
The elliptic curves in class 1710a do not have complex multiplication.Modular form 1710.2.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 Cremona 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 Cremona labels.