Show commands:
SageMath
E = EllipticCurve("gl1")
E.isogeny_class()
Elliptic curves in class 121968gl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.s1 | 121968gl1 | \([0, 0, 0, -1556544, -747468304]\) | \(-78843215872/539\) | \(-2851230659751936\) | \([]\) | \(1728000\) | \(2.1464\) | \(\Gamma_0(N)\)-optimal |
| 121968.s2 | 121968gl2 | \([0, 0, 0, -859584, -1418292304]\) | \(-13278380032/156590819\) | \(-828342382501792198656\) | \([]\) | \(5184000\) | \(2.6957\) | |
| 121968.s3 | 121968gl3 | \([0, 0, 0, 7678176, 36711343856]\) | \(9463555063808/115539436859\) | \(-611186613697316738543616\) | \([]\) | \(15552000\) | \(3.2450\) |
Rank
sage: E.rank()
The elliptic curves in class 121968gl have rank \(1\).
Complex multiplication
The elliptic curves in class 121968gl do not have complex multiplication.Modular form 121968.2.a.gl
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.
