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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 489762.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.q1 | 489762q2 | \([1, -1, 0, -83611683, 223657701385]\) | \(18371176467862382137/4491266165663004\) | \(15803614799417087753518044\) | \([2]\) | \(144506880\) | \(3.5459\) | \(\Gamma_0(N)\)-optimal* |
489762.q2 | 489762q1 | \([1, -1, 0, 12485097, 22065876301]\) | \(61166244013918343/95191642950768\) | \(-334954999732354530158448\) | \([2]\) | \(72253440\) | \(3.1993\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762.q have rank \(1\).
Complex multiplication
The elliptic curves in class 489762.q do not have complex multiplication.Modular form 489762.2.a.q
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.