Show commands:
SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 25350cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25350.cz6 | 25350cs1 | \([1, 0, 0, 63287, 2459417]\) | \(371694959/249600\) | \(-18824555100000000\) | \([2]\) | \(258048\) | \(1.8115\) | \(\Gamma_0(N)\)-optimal |
| 25350.cz5 | 25350cs2 | \([1, 0, 0, -274713, 20373417]\) | \(30400540561/15210000\) | \(1147121326406250000\) | \([2, 2]\) | \(516096\) | \(2.1581\) | |
| 25350.cz3 | 25350cs3 | \([1, 0, 0, -2387213, -1405564083]\) | \(19948814692561/231344100\) | \(17447715374639062500\) | \([2, 2]\) | \(1032192\) | \(2.5047\) | |
| 25350.cz2 | 25350cs4 | \([1, 0, 0, -3570213, 2594158917]\) | \(66730743078481/60937500\) | \(4595838647460937500\) | \([2]\) | \(1032192\) | \(2.5047\) | |
| 25350.cz4 | 25350cs5 | \([1, 0, 0, -485963, -3582495333]\) | \(-168288035761/73415764890\) | \(-5536935542389625156250\) | \([2]\) | \(2064384\) | \(2.8513\) | |
| 25350.cz1 | 25350cs6 | \([1, 0, 0, -38088463, -90480182833]\) | \(81025909800741361/11088090\) | \(836251446950156250\) | \([2]\) | \(2064384\) | \(2.8513\) |
Rank
sage: E.rank()
The elliptic curves in class 25350cs have rank \(1\).
Complex multiplication
The elliptic curves in class 25350cs do not have complex multiplication.Modular form 25350.2.a.cs
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
