Show commands:
SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 121968.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.df1 | 121968bd2 | \([0, 0, 0, -386595, -25227774]\) | \(2415899250/1294139\) | \(3422902407032199168\) | \([2]\) | \(1474560\) | \(2.2476\) | |
121968.df2 | 121968bd1 | \([0, 0, 0, 92565, -3090582]\) | \(66325500/41503\) | \(-54886190200224768\) | \([2]\) | \(737280\) | \(1.9011\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.df have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.df do not have complex multiplication.Modular form 121968.2.a.df
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.