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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 311696cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
311696.cp2 | 311696cp1 | \([0, -1, 0, -27144, 449264]\) | \(304821217/164864\) | \(1196304927555584\) | \([2]\) | \(1344000\) | \(1.5842\) | \(\Gamma_0(N)\)-optimal |
311696.cp1 | 311696cp2 | \([0, -1, 0, -336904, 75287280]\) | \(582810602977/829472\) | \(6018909166764032\) | \([2]\) | \(2688000\) | \(1.9307\) |
Rank
sage: E.rank()
The elliptic curves in class 311696cp have rank \(1\).
Complex multiplication
The elliptic curves in class 311696cp do not have complex multiplication.Modular form 311696.2.a.cp
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.