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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 54450.gn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54450.gn1 | 54450ft2 | \([1, -1, 1, -11705, -497703]\) | \(-128667913/4096\) | \(-5645376000000\) | \([]\) | \(124416\) | \(1.2228\) | |
54450.gn2 | 54450ft1 | \([1, -1, 1, 670, -2703]\) | \(24167/16\) | \(-22052250000\) | \([]\) | \(41472\) | \(0.67351\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54450.gn have rank \(0\).
Complex multiplication
The elliptic curves in class 54450.gn do not have complex multiplication.Modular form 54450.2.a.gn
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.