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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 690.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
690.d1 | 690d2 | \([1, 1, 0, -242, -1554]\) | \(1577505447721/838350\) | \(838350\) | \([2]\) | \(288\) | \(0.086349\) | |
690.d2 | 690d1 | \([1, 1, 0, -12, -36]\) | \(-217081801/285660\) | \(-285660\) | \([2]\) | \(144\) | \(-0.26022\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 690.d have rank \(0\).
Complex multiplication
The elliptic curves in class 690.d do not have complex multiplication.Modular form 690.2.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.