Show commands:
SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 121968.de
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.de1 | 121968cn2 | \([0, 0, 0, -3740715, 2784731994]\) | \(-4904170882875/43904\) | \(-51823420075671552\) | \([]\) | \(2032128\) | \(2.3728\) | |
121968.de2 | 121968cn1 | \([0, 0, 0, -23595, 7547738]\) | \(-897199875/14680064\) | \(-23769628916318208\) | \([]\) | \(677376\) | \(1.8235\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.de have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.de do not have complex multiplication.Modular form 121968.2.a.de
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.