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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 105966.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
105966.r1 | 105966k2 | \([1, -1, 0, -3409992, -538983536]\) | \(10112728515625/5561943408\) | \(2411804390238090498672\) | \([2]\) | \(4300800\) | \(2.7935\) | |
105966.r2 | 105966k1 | \([1, -1, 0, 828648, -66799040]\) | \(145116956375/88397568\) | \(-38331501590274746112\) | \([2]\) | \(2150400\) | \(2.4469\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 105966.r have rank \(0\).
Complex multiplication
The elliptic curves in class 105966.r do not have complex multiplication.Modular form 105966.2.a.r
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.