Show commands:
SageMath
E = EllipticCurve("jr1")
E.isogeny_class()
Elliptic curves in class 476850jr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 476850.jr4 | 476850jr1 | \([1, 0, 0, -135313563, 605830995117]\) | \(726497538898787209/1038579300\) | \(391699679933151562500\) | \([2]\) | \(79626240\) | \(3.2227\) | \(\Gamma_0(N)\)-optimal |
| 476850.jr3 | 476850jr2 | \([1, 0, 0, -136541813, 594271934367]\) | \(746461053445307689/27443694341250\) | \(10350376027762990957031250\) | \([2]\) | \(159252480\) | \(3.5693\) | |
| 476850.jr2 | 476850jr3 | \([1, 0, 0, -172269438, 248960031492]\) | \(1499114720492202169/796539777000000\) | \(300414591071595515625000000\) | \([2]\) | \(238878720\) | \(3.7720\) | |
| 476850.jr1 | 476850jr4 | \([1, 0, 0, -1592126438, -24264871073508]\) | \(1183430669265454849849/10449720703125000\) | \(3941107101600128173828125000\) | \([2]\) | \(477757440\) | \(4.1186\) |
Rank
sage: E.rank()
The elliptic curves in class 476850jr have rank \(1\).
Complex multiplication
The elliptic curves in class 476850jr do not have complex multiplication.Modular form 476850.2.a.jr
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.
