Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 286650j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.j3 | 286650j1 | \([1, -1, 0, -8362692, -4177561284]\) | \(48264326765929/22299191460\) | \(29883049265008229062500\) | \([2]\) | \(31850496\) | \(3.0073\) | \(\Gamma_0(N)\)-optimal |
| 286650.j4 | 286650j2 | \([1, -1, 0, 29453058, -31518348534]\) | \(2108526614950391/1540302022350\) | \(-2064152025397106317968750\) | \([2]\) | \(63700992\) | \(3.3539\) | |
| 286650.j1 | 286650j3 | \([1, -1, 0, -567495567, -5203311064659]\) | \(15082569606665230489/7751016000\) | \(10387102752014625000000\) | \([2]\) | \(95551488\) | \(3.5566\) | |
| 286650.j2 | 286650j4 | \([1, -1, 0, -564408567, -5262720379659]\) | \(-14837772556740428569/342100087875000\) | \(-458446836418716919921875000\) | \([2]\) | \(191102976\) | \(3.9032\) |
Rank
sage: E.rank()
The elliptic curves in class 286650j have rank \(0\).
Complex multiplication
The elliptic curves in class 286650j do not have complex multiplication.Modular form 286650.2.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 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.
