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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 152880ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 152880.ey7 | 152880ca1 | \([0, 1, 0, -20168416, 34668672884]\) | \(1882742462388824401/11650189824000\) | \(5614113515945066496000\) | \([2]\) | \(10616832\) | \(3.0114\) | \(\Gamma_0(N)\)-optimal |
| 152880.ey6 | 152880ca2 | \([0, 1, 0, -32461536, -12600832140]\) | \(7850236389974007121/4400862921000000\) | \(2120733170863017984000000\) | \([2, 2]\) | \(21233664\) | \(3.3580\) | |
| 152880.ey5 | 152880ca3 | \([0, 1, 0, -124597216, -512038109836]\) | \(443915739051786565201/21894701746029840\) | \(10550844480383650390671360\) | \([2]\) | \(31850496\) | \(3.5607\) | |
| 152880.ey8 | 152880ca4 | \([0, 1, 0, 127568544, -99849231756]\) | \(476437916651992691759/284661685546875000\) | \(-137175706185336000000000000\) | \([2]\) | \(42467328\) | \(3.7046\) | |
| 152880.ey4 | 152880ca5 | \([0, 1, 0, -389181536, -2950404064140]\) | \(13527956825588849127121/25701087819771000\) | \(12385105022600144400384000\) | \([2]\) | \(42467328\) | \(3.7046\) | |
| 152880.ey2 | 152880ca6 | \([0, 1, 0, -1969333536, -33638337155340]\) | \(1752803993935029634719121/4599740941532100\) | \(2216570560636149894758400\) | \([2, 2]\) | \(63700992\) | \(3.9073\) | |
| 152880.ey3 | 152880ca7 | \([0, 1, 0, -1945131456, -34505381511756]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-43322654402387488003599360000\) | \([2]\) | \(127401984\) | \(4.2539\) | |
| 152880.ey1 | 152880ca8 | \([0, 1, 0, -31509316736, -2152824915930060]\) | \(7179471593960193209684686321/49441793310\) | \(23825520808461066240\) | \([2]\) | \(127401984\) | \(4.2539\) |
Rank
sage: E.rank()
The elliptic curves in class 152880ca have rank \(2\).
Complex multiplication
The elliptic curves in class 152880ca do not have complex multiplication.Modular form 152880.2.a.ca
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
