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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 429429bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
429429.bi6 | 429429bi1 | \([1, 0, 0, 20023, 241968]\) | \(103823/63\) | \(-538712154467487\) | \([2]\) | \(1474560\) | \(1.5159\) | \(\Gamma_0(N)\)-optimal* |
429429.bi5 | 429429bi2 | \([1, 0, 0, -82222, 1939235]\) | \(7189057/3969\) | \(33938865731451681\) | \([2, 2]\) | \(2949120\) | \(1.8625\) | \(\Gamma_0(N)\)-optimal* |
429429.bi2 | 429429bi3 | \([1, 0, 0, -1002427, 385664720]\) | \(13027640977/21609\) | \(184778268982348041\) | \([2, 2]\) | \(5898240\) | \(2.2091\) | \(\Gamma_0(N)\)-optimal* |
429429.bi3 | 429429bi4 | \([1, 0, 0, -797937, -272752182]\) | \(6570725617/45927\) | \(392721160606798023\) | \([2]\) | \(5898240\) | \(2.2091\) | |
429429.bi1 | 429429bi5 | \([1, 0, 0, -16032442, 24707234993]\) | \(53297461115137/147\) | \(1256995027090803\) | \([2]\) | \(11796480\) | \(2.5556\) | \(\Gamma_0(N)\)-optimal* |
429429.bi4 | 429429bi6 | \([1, 0, 0, -695692, 626206307]\) | \(-4354703137/17294403\) | \(-147884207942205882147\) | \([2]\) | \(11796480\) | \(2.5556\) |
Rank
sage: E.rank()
The elliptic curves in class 429429bi have rank \(1\).
Complex multiplication
The elliptic curves in class 429429bi do not have complex multiplication.Modular form 429429.2.a.bi
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.