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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 11550.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11550.bk1 | 11550bl7 | \([1, 1, 1, -6410688, 5858968281]\) | \(1864737106103260904761/129177711985836360\) | \(2018401749778693125000\) | \([2]\) | \(663552\) | \(2.8360\) | |
| 11550.bk2 | 11550bl4 | \([1, 1, 1, -6300063, 6083844531]\) | \(1769857772964702379561/691787250\) | \(10809175781250\) | \([2]\) | \(221184\) | \(2.2867\) | |
| 11550.bk3 | 11550bl6 | \([1, 1, 1, -1265688, -438511719]\) | \(14351050585434661561/3001282273281600\) | \(46895035520025000000\) | \([2, 2]\) | \(331776\) | \(2.4894\) | |
| 11550.bk4 | 11550bl3 | \([1, 1, 1, -1193688, -502447719]\) | \(12038605770121350841/757333463040\) | \(11833335360000000\) | \([2]\) | \(165888\) | \(2.1429\) | |
| 11550.bk5 | 11550bl2 | \([1, 1, 1, -393813, 94907031]\) | \(432288716775559561/270140062500\) | \(4220938476562500\) | \([2, 2]\) | \(110592\) | \(1.9401\) | |
| 11550.bk6 | 11550bl5 | \([1, 1, 1, -319563, 131883531]\) | \(-230979395175477481/348191894531250\) | \(-5440498352050781250\) | \([2]\) | \(221184\) | \(2.2867\) | |
| 11550.bk7 | 11550bl1 | \([1, 1, 1, -29313, 866031]\) | \(178272935636041/81841914000\) | \(1278779906250000\) | \([2]\) | \(55296\) | \(1.5935\) | \(\Gamma_0(N)\)-optimal |
| 11550.bk8 | 11550bl8 | \([1, 1, 1, 2727312, -2642647719]\) | \(143584693754978072519/276341298967965000\) | \(-4317832796374453125000\) | \([2]\) | \(663552\) | \(2.8360\) |
Rank
sage: E.rank()
The elliptic curves in class 11550.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 11550.bk do not have complex multiplication.Modular form 11550.2.a.bk
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.
