Show commands:
SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 227430.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227430.cf1 | 227430em3 | \([1, -1, 0, -1213569, 514873503]\) | \(5763259856089/5670\) | \(194460855901830\) | \([2]\) | \(3686400\) | \(2.0351\) | |
| 227430.cf2 | 227430em2 | \([1, -1, 0, -76419, 7932033]\) | \(1439069689/44100\) | \(1512473323680900\) | \([2, 2]\) | \(1843200\) | \(1.6885\) | |
| 227430.cf3 | 227430em1 | \([1, -1, 0, -11439, -294435]\) | \(4826809/1680\) | \(57618031378320\) | \([2]\) | \(921600\) | \(1.3420\) | \(\Gamma_0(N)\)-optimal |
| 227430.cf4 | 227430em4 | \([1, -1, 0, 21051, 26704755]\) | \(30080231/9003750\) | \(-308796636918183750\) | \([2]\) | \(3686400\) | \(2.0351\) |
Rank
sage: E.rank()
The elliptic curves in class 227430.cf have rank \(0\).
Complex multiplication
The elliptic curves in class 227430.cf do not have complex multiplication.Modular form 227430.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
