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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 21866.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21866.e1 | 21866c2 | \([1, -1, 0, -178870, -31902586]\) | \(-1064019559329/125497034\) | \(-74648562539529914\) | \([]\) | \(351232\) | \(1.9734\) | |
21866.e2 | 21866c1 | \([1, -1, 0, -2260, 63824]\) | \(-2146689/1664\) | \(-989786006144\) | \([]\) | \(50176\) | \(1.0005\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21866.e have rank \(1\).
Complex multiplication
The elliptic curves in class 21866.e do not have complex multiplication.Modular form 21866.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.