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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2156.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2156.a1 | 2156a2 | \([0, -1, 0, -3789, 91561]\) | \(-199794688/1331\) | \(-40087249664\) | \([]\) | \(2160\) | \(0.86997\) | |
2156.a2 | 2156a1 | \([0, -1, 0, 131, 617]\) | \(8192/11\) | \(-331299584\) | \([]\) | \(720\) | \(0.32066\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2156.a have rank \(1\).
Complex multiplication
The elliptic curves in class 2156.a do not have complex multiplication.Modular form 2156.2.a.a
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.