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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 53312.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.bc1 | 53312s2 | \([0, 0, 0, -25676, -696976]\) | \(60698457/28322\) | \(873478220152832\) | \([2]\) | \(147456\) | \(1.5614\) | |
53312.bc2 | 53312s1 | \([0, 0, 0, 5684, -82320]\) | \(658503/476\) | \(-14680306221056\) | \([2]\) | \(73728\) | \(1.2148\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53312.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 53312.bc do not have complex multiplication.Modular form 53312.2.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 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.