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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 100010.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100010.f1 | 100010e1 | \([1, -1, 1, -17216328052, -869056815249449]\) | \(564345804012377540082892046274202641/313563965869260800000000000000\) | \(313563965869260800000000000000\) | \([2]\) | \(222813696\) | \(4.6074\) | \(\Gamma_0(N)\)-optimal |
100010.f2 | 100010e2 | \([1, -1, 1, -14154486132, -1187958673848361]\) | \(-313621648911503266976083898753158161/428167812500000000000000000000000\) | \(-428167812500000000000000000000000\) | \([2]\) | \(445627392\) | \(4.9540\) |
Rank
sage: E.rank()
The elliptic curves in class 100010.f have rank \(1\).
Complex multiplication
The elliptic curves in class 100010.f do not have complex multiplication.Modular form 100010.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.