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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 70785bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
70785.m1 | 70785bf1 | \([1, -1, 1, -1112, 3314]\) | \(117649/65\) | \(83945417985\) | \([2]\) | \(67200\) | \(0.78664\) | \(\Gamma_0(N)\)-optimal |
70785.m2 | 70785bf2 | \([1, -1, 1, 4333, 22916]\) | \(6967871/4225\) | \(-5456452169025\) | \([2]\) | \(134400\) | \(1.1332\) |
Rank
sage: E.rank()
The elliptic curves in class 70785bf have rank \(1\).
Complex multiplication
The elliptic curves in class 70785bf do not have complex multiplication.Modular form 70785.2.a.bf
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.