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SageMath
E = EllipticCurve("fg1")
E.isogeny_class()
Elliptic curves in class 121968.fg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.fg1 | 121968es2 | \([0, 0, 0, -5877696, -5484635728]\) | \(35084566528/1029\) | \(658634282402697216\) | \([]\) | \(3649536\) | \(2.5189\) | |
121968.fg2 | 121968es1 | \([0, 0, 0, -127776, 5387888]\) | \(360448/189\) | \(120973643706617856\) | \([]\) | \(1216512\) | \(1.9696\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.fg have rank \(0\).
Complex multiplication
The elliptic curves in class 121968.fg do not have complex multiplication.Modular form 121968.2.a.fg
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.