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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 115920cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
115920.a3 | 115920cy1 | \([0, 0, 0, -69123, 398200322]\) | \(-12232183057921/22933241856000\) | \(-68478293250146304000\) | \([2]\) | \(3981312\) | \(2.4851\) | \(\Gamma_0(N)\)-optimal |
115920.a2 | 115920cy2 | \([0, 0, 0, -8547843, 9502649858]\) | \(23131609187144855041/322060536000000\) | \(961667607527424000000\) | \([2]\) | \(7962624\) | \(2.8316\) | |
115920.a4 | 115920cy3 | \([0, 0, 0, 622077, -10748505598]\) | \(8915971454369279/16719623332762560\) | \(-49924527757655679959040\) | \([2]\) | \(11943936\) | \(3.0344\) | |
115920.a1 | 115920cy4 | \([0, 0, 0, -69459843, -218064841342]\) | \(12411881707829361287041/303132494474220600\) | \(905148778380111124070400\) | \([2]\) | \(23887872\) | \(3.3809\) |
Rank
sage: E.rank()
The elliptic curves in class 115920cy have rank \(1\).
Complex multiplication
The elliptic curves in class 115920cy do not have complex multiplication.Modular form 115920.2.a.cy
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.