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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 286110bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.bn3 | 286110bn1 | \([1, -1, 0, -336450, 59320000]\) | \(6462919457883/1414187500\) | \(921646305725062500\) | \([2]\) | \(4644864\) | \(2.1607\) | \(\Gamma_0(N)\)-optimal |
| 286110.bn4 | 286110bn2 | \([1, -1, 0, 747300, 362119750]\) | \(70819203762117/127995282250\) | \(-83416363838563956750\) | \([2]\) | \(9289728\) | \(2.5073\) | |
| 286110.bn1 | 286110bn3 | \([1, -1, 0, -8681325, -9838258075]\) | \(152298969481827/86468800\) | \(41081307046391937600\) | \([2]\) | \(13934592\) | \(2.7100\) | |
| 286110.bn2 | 286110bn4 | \([1, -1, 0, -7120725, -13487877235]\) | \(-84044939142627/116825833960\) | \(-55503926917703987092920\) | \([2]\) | \(27869184\) | \(3.0566\) |
Rank
sage: E.rank()
The elliptic curves in class 286110bn have rank \(1\).
Complex multiplication
The elliptic curves in class 286110bn do not have complex multiplication.Modular form 286110.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
