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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 67270bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67270.x2 | 67270bc1 | \([1, 0, 0, -1446, 21076]\) | \(-11224377919/19600\) | \(-583903600\) | \([2]\) | \(49152\) | \(0.57576\) | \(\Gamma_0(N)\)-optimal |
67270.x1 | 67270bc2 | \([1, 0, 0, -23146, 1353456]\) | \(46032827294719/140\) | \(4170740\) | \([2]\) | \(98304\) | \(0.92233\) |
Rank
sage: E.rank()
The elliptic curves in class 67270bc have rank \(1\).
Complex multiplication
The elliptic curves in class 67270bc do not have complex multiplication.Modular form 67270.2.a.bc
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.