Show commands:
SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 72450.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.dp1 | 72450dn6 | \([1, -1, 1, -17805155, 28922059847]\) | \(54804145548726848737/637608031452\) | \(7262753983257937500\) | \([2]\) | \(4194304\) | \(2.7700\) | |
| 72450.dp2 | 72450dn4 | \([1, -1, 1, -3985655, -3061573153]\) | \(614716917569296417/19093020912\) | \(217481441325750000\) | \([2]\) | \(2097152\) | \(2.4234\) | |
| 72450.dp3 | 72450dn3 | \([1, -1, 1, -1141655, 427474847]\) | \(14447092394873377/1439452851984\) | \(16396267642130250000\) | \([2, 2]\) | \(2097152\) | \(2.4234\) | |
| 72450.dp4 | 72450dn2 | \([1, -1, 1, -259655, -43513153]\) | \(169967019783457/26337394944\) | \(299999389284000000\) | \([2, 2]\) | \(1048576\) | \(2.0769\) | |
| 72450.dp5 | 72450dn1 | \([1, -1, 1, 28345, -3769153]\) | \(221115865823/664731648\) | \(-7571708928000000\) | \([2]\) | \(524288\) | \(1.7303\) | \(\Gamma_0(N)\)-optimal |
| 72450.dp6 | 72450dn5 | \([1, -1, 1, 1409845, 2065537847]\) | \(27207619911317663/177609314617308\) | \(-2023081099312773937500\) | \([2]\) | \(4194304\) | \(2.7700\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.dp have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.dp do not have complex multiplication.Modular form 72450.2.a.dp
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
