Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 25350k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25350.g4 | 25350k1 | \([1, 1, 0, 204825, -14686875]\) | \(5735339/3888\) | \(-644223336909750000\) | \([2]\) | \(399360\) | \(2.1059\) | \(\Gamma_0(N)\)-optimal |
| 25350.g3 | 25350k2 | \([1, 1, 0, -893675, -123438375]\) | \(476379541/236196\) | \(39136567717267312500\) | \([2]\) | \(798720\) | \(2.4525\) | |
| 25350.g2 | 25350k3 | \([1, 1, 0, -18744300, 31247250000]\) | \(-4395631034341/3145728\) | \(-521232353181696000000\) | \([2]\) | \(1996800\) | \(2.9106\) | |
| 25350.g1 | 25350k4 | \([1, 1, 0, -299960300, 1999478034000]\) | \(18013780041269221/9216\) | \(1527047909712000000\) | \([2]\) | \(3993600\) | \(3.2572\) |
Rank
sage: E.rank()
The elliptic curves in class 25350k have rank \(0\).
Complex multiplication
The elliptic curves in class 25350k do not have complex multiplication.Modular form 25350.2.a.k
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.
