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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 12696.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12696.k1 | 12696j5 | \([0, -1, 0, -203312, 35352972]\) | \(3065617154/9\) | \(2728597506048\) | \([2]\) | \(45056\) | \(1.6155\) | |
| 12696.k2 | 12696j3 | \([0, -1, 0, -34032, -2404932]\) | \(28756228/3\) | \(454766251008\) | \([2]\) | \(22528\) | \(1.2690\) | |
| 12696.k3 | 12696j4 | \([0, -1, 0, -12872, 540540]\) | \(1556068/81\) | \(12278688777216\) | \([2, 2]\) | \(22528\) | \(1.2690\) | |
| 12696.k4 | 12696j2 | \([0, -1, 0, -2292, -30780]\) | \(35152/9\) | \(341074688256\) | \([2, 2]\) | \(11264\) | \(0.92239\) | |
| 12696.k5 | 12696j1 | \([0, -1, 0, 353, -3272]\) | \(2048/3\) | \(-7105722672\) | \([2]\) | \(5632\) | \(0.57582\) | \(\Gamma_0(N)\)-optimal |
| 12696.k6 | 12696j6 | \([0, -1, 0, 8288, 2123308]\) | \(207646/6561\) | \(-1989147581908992\) | \([2]\) | \(45056\) | \(1.6155\) |
Rank
sage: E.rank()
The elliptic curves in class 12696.k have rank \(1\).
Complex multiplication
The elliptic curves in class 12696.k do not have complex multiplication.Modular form 12696.2.a.k
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
