Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 29744.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29744.k1 | 29744bb2 | \([0, -1, 0, -282117, 59071609]\) | \(-2009615368192/53094899\) | \(-65607407704906496\) | \([]\) | \(241920\) | \(2.0094\) | |
| 29744.k2 | 29744bb1 | \([0, -1, 0, 15323, 327209]\) | \(321978368/224939\) | \(-277948822950656\) | \([]\) | \(80640\) | \(1.4601\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 29744.k have rank \(1\).
Complex multiplication
The elliptic curves in class 29744.k do not have complex multiplication.Modular form 29744.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
