Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 6498.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6498.p1 | 6498w3 | \([1, -1, 1, -284456516, -1846525107769]\) | \(74220219816682217473/16416\) | \(563010478039584\) | \([2]\) | \(691200\) | \(3.1230\) | |
| 6498.p2 | 6498w2 | \([1, -1, 1, -17778596, -28848405049]\) | \(18120364883707393/269485056\) | \(9242380007497810944\) | \([2, 2]\) | \(345600\) | \(2.7764\) | |
| 6498.p3 | 6498w4 | \([1, -1, 1, -17258756, -30615029305]\) | \(-16576888679672833/2216253521952\) | \(-76009622006037231510048\) | \([2]\) | \(691200\) | \(3.1230\) | |
| 6498.p4 | 6498w1 | \([1, -1, 1, -1143716, -422722105]\) | \(4824238966273/537919488\) | \(18448727344401088512\) | \([4]\) | \(172800\) | \(2.4298\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6498.p have rank \(0\).
Complex multiplication
The elliptic curves in class 6498.p do not have complex multiplication.Modular form 6498.2.a.p
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.
