Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 457776e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
457776.e3 | 457776e1 | \([0, 0, 0, -1873587, 719508850]\) | \(10091699281/2737152\) | \(197278573965650952192\) | \([2]\) | \(19660800\) | \(2.6022\) | \(\Gamma_0(N)\)-optimal |
457776.e4 | 457776e2 | \([0, 0, 0, 4784973, 4681352050]\) | \(168105213359/228637728\) | \(-16478925881568281100288\) | \([2]\) | \(39321600\) | \(2.9488\) | |
457776.e1 | 457776e3 | \([0, 0, 0, -418865907, -3299594922830]\) | \(112763292123580561/1932612\) | \(139291840346792804352\) | \([2]\) | \(98304000\) | \(3.4069\) | |
457776.e2 | 457776e4 | \([0, 0, 0, -418449747, -3306478625390]\) | \(-112427521449300721/466873642818\) | \(-33649635269536991900540928\) | \([2]\) | \(196608000\) | \(3.7535\) |
Rank
sage: E.rank()
The elliptic curves in class 457776e have rank \(1\).
Complex multiplication
The elliptic curves in class 457776e do not have complex multiplication.Modular form 457776.2.a.e
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.