Show commands:
SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 121968ez
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121968.d2 | 121968ez1 | \([0, 0, 0, -1452, 86515]\) | \(-16384/147\) | \(-3037532663088\) | \([2]\) | \(268800\) | \(1.0777\) | \(\Gamma_0(N)\)-optimal |
| 121968.d1 | 121968ez2 | \([0, 0, 0, -39567, 3021370]\) | \(20720464/63\) | \(20828795404032\) | \([2]\) | \(537600\) | \(1.4243\) |
Rank
sage: E.rank()
The elliptic curves in class 121968ez have rank \(0\).
Complex multiplication
The elliptic curves in class 121968ez do not have complex multiplication.Modular form 121968.2.a.ez
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
