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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 198198cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 198198.d3 | 198198cp1 | \([1, -1, 0, 14679, 6505461]\) | \(270840023/14329224\) | \(-18505733816626056\) | \([]\) | \(2332800\) | \(1.8011\) | \(\Gamma_0(N)\)-optimal |
| 198198.d2 | 198198cp2 | \([1, -1, 0, -132336, -177410304]\) | \(-198461344537/10417365504\) | \(-13453693869781541376\) | \([]\) | \(6998400\) | \(2.3504\) | |
| 198198.d1 | 198198cp3 | \([1, -1, 0, -28375551, -58172622579]\) | \(-1956469094246217097/36641439744\) | \(-47321245767419559936\) | \([]\) | \(20995200\) | \(2.8997\) |
Rank
sage: E.rank()
The elliptic curves in class 198198cp have rank \(1\).
Complex multiplication
The elliptic curves in class 198198cp do not have complex multiplication.Modular form 198198.2.a.cp
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
