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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 491985bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
491985.bc1 | 491985bc1 | \([0, 0, 1, -35322, 2566942]\) | \(-303464448/1625\) | \(-26097873208875\) | \([]\) | \(1161216\) | \(1.4189\) | \(\Gamma_0(N)\)-optimal |
491985.bc2 | 491985bc2 | \([0, 0, 1, 90828, 13663937]\) | \(7077888/10985\) | \(-128611363088264355\) | \([]\) | \(3483648\) | \(1.9682\) |
Rank
sage: E.rank()
The elliptic curves in class 491985bc have rank \(1\).
Complex multiplication
The elliptic curves in class 491985bc do not have complex multiplication.Modular form 491985.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.