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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 177744.dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177744.dd1 | 177744x6 | \([0, 1, 0, -669790352, -6672162455532]\) | \(54804145548726848737/637608031452\) | \(386616818768022654271488\) | \([2]\) | \(51904512\) | \(3.6769\) | |
| 177744.dd2 | 177744x3 | \([0, 1, 0, -149931472, 706553332628]\) | \(614716917569296417/19093020912\) | \(11577148720756780105728\) | \([4]\) | \(25952256\) | \(3.3303\) | |
| 177744.dd3 | 177744x4 | \([0, 1, 0, -42946512, -98576474220]\) | \(14447092394873377/1439452851984\) | \(872819435999382953066496\) | \([2, 2]\) | \(25952256\) | \(3.3303\) | |
| 177744.dd4 | 177744x2 | \([0, 1, 0, -9767632, 10051178900]\) | \(169967019783457/26337394944\) | \(15969811146666578804736\) | \([2, 2]\) | \(12976128\) | \(2.9837\) | |
| 177744.dd5 | 177744x1 | \([0, 1, 0, 1066288, 868348308]\) | \(221115865823/664731648\) | \(-403063359316439334912\) | \([2]\) | \(6488064\) | \(2.6371\) | \(\Gamma_0(N)\)-optimal |
| 177744.dd6 | 177744x5 | \([0, 1, 0, 53035248, -476629430508]\) | \(27207619911317663/177609314617308\) | \(-107694296203484711185661952\) | \([2]\) | \(51904512\) | \(3.6769\) |
Rank
sage: E.rank()
The elliptic curves in class 177744.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 177744.dd do not have complex multiplication.Modular form 177744.2.a.dd
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.
