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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 8400.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8400.e1 | 8400bj2 | \([0, -1, 0, -908, 10812]\) | \(20720464/63\) | \(252000000\) | \([2]\) | \(3840\) | \(0.48073\) | |
8400.e2 | 8400bj1 | \([0, -1, 0, -33, 312]\) | \(-16384/147\) | \(-36750000\) | \([2]\) | \(1920\) | \(0.13416\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8400.e have rank \(0\).
Complex multiplication
The elliptic curves in class 8400.e do not have complex multiplication.Modular form 8400.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.