Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 66654.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 66654.j1 | 66654w4 | \([1, -1, 0, -6398883, 6231834121]\) | \(268498407453697/252\) | \(27195377096412\) | \([2]\) | \(1441792\) | \(2.3062\) | |
| 66654.j2 | 66654w6 | \([1, -1, 0, -4351653, -3459419429]\) | \(84448510979617/933897762\) | \(100784530980496924722\) | \([2]\) | \(2883584\) | \(2.6527\) | |
| 66654.j3 | 66654w3 | \([1, -1, 0, -495243, 47599825]\) | \(124475734657/63011844\) | \(6800122456826531364\) | \([2, 2]\) | \(1441792\) | \(2.3062\) | |
| 66654.j4 | 66654w2 | \([1, -1, 0, -400023, 97399885]\) | \(65597103937/63504\) | \(6853235028295824\) | \([2, 2]\) | \(720896\) | \(1.9596\) | |
| 66654.j5 | 66654w1 | \([1, -1, 0, -19143, 2256061]\) | \(-7189057/16128\) | \(-1740504134170368\) | \([2]\) | \(360448\) | \(1.6130\) | \(\Gamma_0(N)\)-optimal |
| 66654.j6 | 66654w5 | \([1, -1, 0, 1837647, 366272599]\) | \(6359387729183/4218578658\) | \(-455261259584070125298\) | \([2]\) | \(2883584\) | \(2.6527\) |
Rank
sage: E.rank()
The elliptic curves in class 66654.j have rank \(0\).
Complex multiplication
The elliptic curves in class 66654.j do not have complex multiplication.Modular form 66654.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
