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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 105710.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
105710.f1 | 105710i1 | \([1, 1, 1, -981, -32717]\) | \(-117649/440\) | \(-390501619640\) | \([]\) | \(119880\) | \(0.90989\) | \(\Gamma_0(N)\)-optimal |
105710.f2 | 105710i2 | \([1, 1, 1, 8629, 770679]\) | \(80062991/332750\) | \(-295316849852750\) | \([]\) | \(359640\) | \(1.4592\) |
Rank
sage: E.rank()
The elliptic curves in class 105710.f have rank \(0\).
Complex multiplication
The elliptic curves in class 105710.f do not have complex multiplication.Modular form 105710.2.a.f
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.