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SageMath
E = EllipticCurve("hp1")
E.isogeny_class()
Elliptic curves in class 476850.hp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.hp1 | 476850hp2 | \([1, 1, 1, -1341700713, -18916657980969]\) | \(708234550511150304361/23696640000\) | \(8937176297940000000000\) | \([2]\) | \(194641920\) | \(3.7116\) | |
476850.hp2 | 476850hp1 | \([1, 1, 1, -83972713, -294737212969]\) | \(173629978755828841/1000026931200\) | \(377159672714035200000000\) | \([2]\) | \(97320960\) | \(3.3651\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 476850.hp have rank \(1\).
Complex multiplication
The elliptic curves in class 476850.hp do not have complex multiplication.Modular form 476850.2.a.hp
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.