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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 67270c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67270.q2 | 67270c1 | \([1, 1, 0, 11032, -331828]\) | \(167284151/151900\) | \(-134811809143900\) | \([2]\) | \(307200\) | \(1.3986\) | \(\Gamma_0(N)\)-optimal |
67270.q1 | 67270c2 | \([1, 1, 0, -56238, -3036082]\) | \(22164361129/8408750\) | \(7462796577608750\) | \([2]\) | \(614400\) | \(1.7451\) |
Rank
sage: E.rank()
The elliptic curves in class 67270c have rank \(2\).
Complex multiplication
The elliptic curves in class 67270c do not have complex multiplication.Modular form 67270.2.a.c
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.