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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 414736.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414736.e1 | 414736e2 | \([0, 1, 0, -47495912, -126026969804]\) | \(-313994137/64\) | \(-2415187488230439387136\) | \([]\) | \(47692800\) | \(3.1005\) | |
414736.e2 | 414736e1 | \([0, 1, 0, 198728, -590066604]\) | \(23/4\) | \(-150949218014402461696\) | \([]\) | \(15897600\) | \(2.5512\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414736.e have rank \(0\).
Complex multiplication
The elliptic curves in class 414736.e do not have complex multiplication.Modular form 414736.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.