Show commands:
SageMath
E = EllipticCurve("jd1")
E.isogeny_class()
Elliptic curves in class 142800.jd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 142800.jd1 | 142800by5 | \([0, 1, 0, -28696008, 59042807988]\) | \(40832710302042509761/91556816413125\) | \(5859636250440000000000\) | \([2]\) | \(12582912\) | \(3.0591\) | |
| 142800.jd2 | 142800by3 | \([0, 1, 0, -2446008, 190307988]\) | \(25288177725059761/14387797265625\) | \(920819025000000000000\) | \([2, 2]\) | \(6291456\) | \(2.7125\) | |
| 142800.jd3 | 142800by2 | \([0, 1, 0, -1564008, -749904012]\) | \(6610905152742241/35128130625\) | \(2248200360000000000\) | \([2, 2]\) | \(3145728\) | \(2.3659\) | |
| 142800.jd4 | 142800by1 | \([0, 1, 0, -1562008, -751924012]\) | \(6585576176607121/187425\) | \(11995200000000\) | \([2]\) | \(1572864\) | \(2.0194\) | \(\Gamma_0(N)\)-optimal |
| 142800.jd5 | 142800by4 | \([0, 1, 0, -714008, -1560804012]\) | \(-629004249876241/16074715228425\) | \(-1028781774619200000000\) | \([2]\) | \(6291456\) | \(2.7125\) | |
| 142800.jd6 | 142800by6 | \([0, 1, 0, 9691992, 1525487988]\) | \(1573196002879828319/926055908203125\) | \(-59267578125000000000000\) | \([2]\) | \(12582912\) | \(3.0591\) |
Rank
sage: E.rank()
The elliptic curves in class 142800.jd have rank \(0\).
Complex multiplication
The elliptic curves in class 142800.jd do not have complex multiplication.Modular form 142800.2.a.jd
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
