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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 23232ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
23232.cj2 | 23232ce1 | \([0, 1, 0, -37429, -3252229]\) | \(-3196715008/649539\) | \(-1178314711428096\) | \([2]\) | \(115200\) | \(1.6143\) | \(\Gamma_0(N)\)-optimal |
23232.cj1 | 23232ce2 | \([0, 1, 0, -625489, -190608145]\) | \(932410994128/29403\) | \(853429585231872\) | \([2]\) | \(230400\) | \(1.9609\) |
Rank
sage: E.rank()
The elliptic curves in class 23232ce have rank \(1\).
Complex multiplication
The elliptic curves in class 23232ce do not have complex multiplication.Modular form 23232.2.a.ce
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.