Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 232845.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232845.e1 | 232845e4 | \([1, 0, 0, -248556, -47716905]\) | \(36097320816649/80625\) | \(3793074155625\) | \([2]\) | \(1267200\) | \(1.6589\) | |
| 232845.e2 | 232845e3 | \([1, 0, 0, -42786, 2449821]\) | \(184122897769/51282015\) | \(2412607575130215\) | \([2]\) | \(1267200\) | \(1.6589\) | |
| 232845.e3 | 232845e2 | \([1, 0, 0, -15711, -728784]\) | \(9116230969/416025\) | \(19572262643025\) | \([2, 2]\) | \(633600\) | \(1.3123\) | |
| 232845.e4 | 232845e1 | \([1, 0, 0, 534, -43245]\) | \(357911/17415\) | \(-819304017615\) | \([2]\) | \(316800\) | \(0.96572\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 232845.e have rank \(0\).
Complex multiplication
The elliptic curves in class 232845.e do not have complex multiplication.Modular form 232845.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
