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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 9408q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9408.h5 | 9408q1 | \([0, -1, 0, 131, -755]\) | \(2048/3\) | \(-361417728\) | \([2]\) | \(3072\) | \(0.32760\) | \(\Gamma_0(N)\)-optimal |
| 9408.h4 | 9408q2 | \([0, -1, 0, -849, -6831]\) | \(35152/9\) | \(17348050944\) | \([2, 2]\) | \(6144\) | \(0.67418\) | |
| 9408.h2 | 9408q3 | \([0, -1, 0, -12609, -540735]\) | \(28756228/3\) | \(23130734592\) | \([2]\) | \(12288\) | \(1.0208\) | |
| 9408.h3 | 9408q4 | \([0, -1, 0, -4769, 122529]\) | \(1556068/81\) | \(624529833984\) | \([2, 2]\) | \(12288\) | \(1.0208\) | |
| 9408.h1 | 9408q5 | \([0, -1, 0, -75329, 7982913]\) | \(3065617154/9\) | \(138784407552\) | \([2]\) | \(24576\) | \(1.3673\) | |
| 9408.h6 | 9408q6 | \([0, -1, 0, 3071, 478465]\) | \(207646/6561\) | \(-101173833105408\) | \([2]\) | \(24576\) | \(1.3673\) |
Rank
sage: E.rank()
The elliptic curves in class 9408q have rank \(2\).
Complex multiplication
The elliptic curves in class 9408q do not have complex multiplication.Modular form 9408.2.a.q
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
