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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 354900.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
354900.r1 | 354900r2 | \([0, -1, 0, -243986708, -1466318018088]\) | \(665567485783184/257298363\) | \(620965027106833500000000\) | \([2]\) | \(77414400\) | \(3.5308\) | |
354900.r2 | 354900r1 | \([0, -1, 0, -12984833, -29948359338]\) | \(-1605176213504/1640558367\) | \(-247458184089403218750000\) | \([2]\) | \(38707200\) | \(3.1843\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 354900.r have rank \(1\).
Complex multiplication
The elliptic curves in class 354900.r do not have complex multiplication.Modular form 354900.2.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 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.