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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 12168.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12168.j1 | 12168q5 | \([0, 0, 0, -584571, 172029494]\) | \(3065617154/9\) | \(64857485002752\) | \([2]\) | \(73728\) | \(1.8796\) | |
| 12168.j2 | 12168q3 | \([0, 0, 0, -97851, -11780314]\) | \(28756228/3\) | \(10809580833792\) | \([2]\) | \(36864\) | \(1.5330\) | |
| 12168.j3 | 12168q4 | \([0, 0, 0, -37011, 2614430]\) | \(1556068/81\) | \(291858682512384\) | \([2, 2]\) | \(36864\) | \(1.5330\) | |
| 12168.j4 | 12168q2 | \([0, 0, 0, -6591, -153790]\) | \(35152/9\) | \(8107185625344\) | \([2, 2]\) | \(18432\) | \(1.1864\) | |
| 12168.j5 | 12168q1 | \([0, 0, 0, 1014, -15379]\) | \(2048/3\) | \(-168899700528\) | \([2]\) | \(9216\) | \(0.83986\) | \(\Gamma_0(N)\)-optimal |
| 12168.j6 | 12168q6 | \([0, 0, 0, 23829, 10365446]\) | \(207646/6561\) | \(-47281106567006208\) | \([2]\) | \(73728\) | \(1.8796\) |
Rank
sage: E.rank()
The elliptic curves in class 12168.j have rank \(1\).
Complex multiplication
The elliptic curves in class 12168.j do not have complex multiplication.Modular form 12168.2.a.j
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.
