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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 489762.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.m1 | 489762m1 | \([1, -1, 0, -1639923, -806931839]\) | \(1703492778807/2384732\) | \(682800002966871972\) | \([2]\) | \(11261952\) | \(2.3258\) | \(\Gamma_0(N)\)-optimal |
489762.m2 | 489762m2 | \([1, -1, 0, -1178553, -1270977785]\) | \(-632291491287/2072502446\) | \(-593401974006995091666\) | \([2]\) | \(22523904\) | \(2.6724\) |
Rank
sage: E.rank()
The elliptic curves in class 489762.m have rank \(2\).
Complex multiplication
The elliptic curves in class 489762.m do not have complex multiplication.Modular form 489762.2.a.m
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.