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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 159936.gm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.gm1 | 159936m2 | \([0, 1, 0, -1196449, 503249375]\) | \(6141556990297/1019592\) | \(31445215925501952\) | \([2]\) | \(1769472\) | \(2.1736\) | |
159936.gm2 | 159936m1 | \([0, 1, 0, -67489, 9442271]\) | \(-1102302937/616896\) | \(-19025676862488576\) | \([2]\) | \(884736\) | \(1.8270\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 159936.gm have rank \(0\).
Complex multiplication
The elliptic curves in class 159936.gm do not have complex multiplication.Modular form 159936.2.a.gm
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.