Show commands for:
SageMath
sage: E = EllipticCurve("gr1")
sage: E.isogeny_class()
Elliptic curves in class 127050.gr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 127050.gr1 | 127050gb8 | [1, 1, 1, -19514338, -20882966719] | [2] | 19906560 | |
| 127050.gr2 | 127050gb5 | [1, 1, 1, -17427088, -28009019719] | [2] | 6635520 | |
| 127050.gr3 | 127050gb6 | [1, 1, 1, -8170588, 8746908281] | [2, 2] | 9953280 | |
| 127050.gr4 | 127050gb3 | [1, 1, 1, -8110088, 8886300281] | [2] | 4976640 | |
| 127050.gr5 | 127050gb2 | [1, 1, 1, -1092088, -435539719] | [2, 2] | 3317760 | |
| 127050.gr6 | 127050gb4 | [1, 1, 1, -245088, -1092811719] | [2] | 6635520 | |
| 127050.gr7 | 127050gb1 | [1, 1, 1, -124088, 5868281] | [2] | 1658880 | \(\Gamma_0(N)\)-optimal |
| 127050.gr8 | 127050gb7 | [1, 1, 1, 2205162, 29456905281] | [2] | 19906560 |
Rank
sage: E.rank()
The elliptic curves in class 127050.gr have rank \(0\).
Complex multiplication
The elliptic curves in class 127050.gr do not have complex multiplication.Modular form 127050.2.a.gr
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
