Show commands:
SageMath
E = EllipticCurve("ia1")
E.isogeny_class()
Elliptic curves in class 43200ia
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 43200.eo3 | 43200ia1 | \([0, 0, 0, 2100, 26000]\) | \(9261/8\) | \(-884736000000\) | \([]\) | \(41472\) | \(0.97993\) | \(\Gamma_0(N)\)-optimal |
| 43200.eo2 | 43200ia2 | \([0, 0, 0, -21900, -1654000]\) | \(-1167051/512\) | \(-509607936000000\) | \([]\) | \(124416\) | \(1.5292\) | |
| 43200.eo1 | 43200ia3 | \([0, 0, 0, -45900, 3834000]\) | \(-132651/2\) | \(-161243136000000\) | \([]\) | \(124416\) | \(1.5292\) |
Rank
sage: E.rank()
The elliptic curves in class 43200ia have rank \(1\).
Complex multiplication
The elliptic curves in class 43200ia do not have complex multiplication.Modular form 43200.2.a.ia
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.
\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
