Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5780e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5780.f3 | 5780e1 | \([0, -1, 0, -385, 2130]\) | \(16384/5\) | \(1931005520\) | \([2]\) | \(2304\) | \(0.48666\) | \(\Gamma_0(N)\)-optimal |
| 5780.f4 | 5780e2 | \([0, -1, 0, 1060, 13112]\) | \(21296/25\) | \(-154480441600\) | \([2]\) | \(4608\) | \(0.83323\) | |
| 5780.f1 | 5780e3 | \([0, -1, 0, -11945, -498418]\) | \(488095744/125\) | \(48275138000\) | \([2]\) | \(6912\) | \(1.0360\) | |
| 5780.f2 | 5780e4 | \([0, -1, 0, -10500, -625000]\) | \(-20720464/15625\) | \(-96550276000000\) | \([2]\) | \(13824\) | \(1.3825\) |
Rank
sage: E.rank()
The elliptic curves in class 5780e have rank \(1\).
Complex multiplication
The elliptic curves in class 5780e do not have complex multiplication.Modular form 5780.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
