Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 98022.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 98022.d1 | 98022e3 | \([1, 1, 0, -721250, 177945204]\) | \(46753267515625/11591221248\) | \(10287251524885413888\) | \([2]\) | \(1995840\) | \(2.3584\) | |
| 98022.d2 | 98022e1 | \([1, 1, 0, -245555, -46920147]\) | \(1845026709625/793152\) | \(703925319592512\) | \([2]\) | \(665280\) | \(1.8091\) | \(\Gamma_0(N)\)-optimal |
| 98022.d3 | 98022e2 | \([1, 1, 0, -207115, -62057819]\) | \(-1107111813625/1228691592\) | \(-1090468310713750152\) | \([2]\) | \(1330560\) | \(2.1557\) | |
| 98022.d4 | 98022e4 | \([1, 1, 0, 1738910, 1131011188]\) | \(655215969476375/1001033261568\) | \(-888420704445035831808\) | \([2]\) | \(3991680\) | \(2.7050\) |
Rank
sage: E.rank()
The elliptic curves in class 98022.d have rank \(1\).
Complex multiplication
The elliptic curves in class 98022.d do not have complex multiplication.Modular form 98022.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
