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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 121968.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.r1 | 121968gi2 | \([0, 0, 0, -59169, 9808139]\) | \(-1108671232/1369599\) | \(-28300691821990896\) | \([]\) | \(829440\) | \(1.8503\) | |
121968.r2 | 121968gi1 | \([0, 0, 0, 6171, -254221]\) | \(1257728/2079\) | \(-42959390520816\) | \([]\) | \(276480\) | \(1.3010\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.r have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.r do not have complex multiplication.Modular form 121968.2.a.r
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.