Show commands:
SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 25200ea
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.o3 | 25200ea1 | \([0, 0, 0, -12675, -346750]\) | \(4826809/1680\) | \(78382080000000\) | \([2]\) | \(73728\) | \(1.3676\) | \(\Gamma_0(N)\)-optimal |
| 25200.o2 | 25200ea2 | \([0, 0, 0, -84675, 9229250]\) | \(1439069689/44100\) | \(2057529600000000\) | \([2, 2]\) | \(147456\) | \(1.7142\) | |
| 25200.o4 | 25200ea3 | \([0, 0, 0, 23325, 31153250]\) | \(30080231/9003750\) | \(-420078960000000000\) | \([2]\) | \(294912\) | \(2.0608\) | |
| 25200.o1 | 25200ea4 | \([0, 0, 0, -1344675, 600169250]\) | \(5763259856089/5670\) | \(264539520000000\) | \([2]\) | \(294912\) | \(2.0608\) |
Rank
sage: E.rank()
The elliptic curves in class 25200ea have rank \(1\).
Complex multiplication
The elliptic curves in class 25200ea do not have complex multiplication.Modular form 25200.2.a.ea
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
