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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 10710.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10710.d1 | 10710b2 | \([1, -1, 0, -197520, 33837596]\) | \(43297905398453523/9912700\) | \(195111674100\) | \([2]\) | \(59904\) | \(1.5473\) | |
10710.d2 | 10710b1 | \([1, -1, 0, -12300, 535040]\) | \(-10456049121363/160002640\) | \(-3149331963120\) | \([2]\) | \(29952\) | \(1.2007\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10710.d have rank \(1\).
Complex multiplication
The elliptic curves in class 10710.d do not have complex multiplication.Modular form 10710.2.a.d
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.