Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 86640.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86640.j1 | 86640bw4 | \([0, -1, 0, -3587016, -2609861520]\) | \(26487576322129/44531250\) | \(8581168694400000000\) | \([2]\) | \(2211840\) | \(2.5283\) | |
| 86640.j2 | 86640bw2 | \([0, -1, 0, -294696, -12879504]\) | \(14688124849/8122500\) | \(1565205169858560000\) | \([2, 2]\) | \(1105920\) | \(2.1817\) | |
| 86640.j3 | 86640bw1 | \([0, -1, 0, -179176, 29077360]\) | \(3301293169/22800\) | \(4393558371532800\) | \([2]\) | \(552960\) | \(1.8351\) | \(\Gamma_0(N)\)-optimal |
| 86640.j4 | 86640bw3 | \([0, -1, 0, 1149304, -102985104]\) | \(871257511151/527800050\) | \(-101707031937409228800\) | \([2]\) | \(2211840\) | \(2.5283\) |
Rank
sage: E.rank()
The elliptic curves in class 86640.j have rank \(1\).
Complex multiplication
The elliptic curves in class 86640.j do not have complex multiplication.Modular form 86640.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
