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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 417450r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
417450.r1 | 417450r1 | \([1, 1, 0, -37283651000, 2770915804950000]\) | \(275601091196478935659903044731/104123070000\) | \(2165434471406250000\) | \([2]\) | \(464486400\) | \(4.2605\) | \(\Gamma_0(N)\)-optimal |
417450.r2 | 417450r2 | \([1, 1, 0, -37283645500, 2770916663351500]\) | \(-275600969228345132090733365051/169400214159764062500\) | \(-3522995078853843237304687500\) | \([2]\) | \(928972800\) | \(4.6071\) |
Rank
sage: E.rank()
The elliptic curves in class 417450r have rank \(1\).
Complex multiplication
The elliptic curves in class 417450r do not have complex multiplication.Modular form 417450.2.a.r
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 Cremona 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 Cremona labels.