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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5010d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5010.c2 | 5010d1 | \([1, 0, 1, 716, 67646]\) | \(40675641638471/1996889557500\) | \(-1996889557500\) | \([2]\) | \(10752\) | \(1.0400\) | \(\Gamma_0(N)\)-optimal |
5010.c1 | 5010d2 | \([1, 0, 1, -21154, 1134902]\) | \(1046819248735488409/47650971093750\) | \(47650971093750\) | \([2]\) | \(21504\) | \(1.3865\) |
Rank
sage: E.rank()
The elliptic curves in class 5010d have rank \(1\).
Complex multiplication
The elliptic curves in class 5010d do not have complex multiplication.Modular form 5010.2.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 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.