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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 32490bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32490.bf4 | 32490bn1 | \([1, -1, 1, 30329347, -136709816763]\) | \(89962967236397039/287450726400000\) | \(-9858538674664331673600000\) | \([2]\) | \(6912000\) | \(3.4791\) | \(\Gamma_0(N)\)-optimal |
| 32490.bf3 | 32490bn2 | \([1, -1, 1, -285733373, -1601597311419]\) | \(75224183150104868881/11219310000000000\) | \(384782473585178190000000000\) | \([2]\) | \(13824000\) | \(3.8256\) | |
| 32490.bf2 | 32490bn3 | \([1, -1, 1, -10726459853, -427592566016283]\) | \(-3979640234041473454886161/1471455901872240\) | \(-50465709717790999497467760\) | \([2]\) | \(34560000\) | \(4.2838\) | |
| 32490.bf1 | 32490bn4 | \([1, -1, 1, -171623372873, -27366047842833219]\) | \(16300610738133468173382620881/2228489100\) | \(76429258863121485900\) | \([2]\) | \(69120000\) | \(4.6304\) |
Rank
sage: E.rank()
The elliptic curves in class 32490bn have rank \(0\).
Complex multiplication
The elliptic curves in class 32490bn do not have complex multiplication.Modular form 32490.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
