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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 49686v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49686.s3 | 49686v1 | \([1, 1, 0, -3743184, 2782381932]\) | \(4649101309/6804\) | \(8488729912778659908\) | \([2]\) | \(1497600\) | \(2.5338\) | \(\Gamma_0(N)\)-optimal |
49686.s4 | 49686v2 | \([1, 1, 0, -2666654, 4417200390]\) | \(-1680914269/5786802\) | \(-7219664790818250251754\) | \([2]\) | \(2995200\) | \(2.8803\) | |
49686.s1 | 49686v3 | \([1, 1, 0, -111934449, -455703455787]\) | \(124318741396429/51631104\) | \(64415416953936789931008\) | \([2]\) | \(7488000\) | \(3.3385\) | |
49686.s2 | 49686v4 | \([1, 1, 0, -94709969, -600709463115]\) | \(-75306487574989/81352871712\) | \(-101496554319826465159729824\) | \([2]\) | \(14976000\) | \(3.6851\) |
Rank
sage: E.rank()
The elliptic curves in class 49686v have rank \(1\).
Complex multiplication
The elliptic curves in class 49686v do not have complex multiplication.Modular form 49686.2.a.v
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.