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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 309168.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
309168.d1 | 309168d1 | \([0, 0, 0, -71263227, -136593274070]\) | \(13403946614821979039929/5057590268826067968\) | \(15101883621270337735360512\) | \([2]\) | \(121405440\) | \(3.5310\) | \(\Gamma_0(N)\)-optimal |
309168.d2 | 309168d2 | \([0, 0, 0, 222934533, -974762692310]\) | \(410363075617640914325831/374944243169850027552\) | \(-1119577510997281464669831168\) | \([2]\) | \(242810880\) | \(3.8775\) |
Rank
sage: E.rank()
The elliptic curves in class 309168.d have rank \(1\).
Complex multiplication
The elliptic curves in class 309168.d do not have complex multiplication.Modular form 309168.2.a.d
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.