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SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 19110.bq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.bq1 | 19110bo7 | \([1, 1, 1, -1969332296, 33636904645259]\) | \(7179471593960193209684686321/49441793310\) | \(5816777541128190\) | \([2]\) | \(5308416\) | \(3.5607\) | |
| 19110.bq2 | 19110bo6 | \([1, 1, 1, -123083346, 525537476379]\) | \(1752803993935029634719121/4599740941532100\) | \(541154922030310032900\) | \([2, 2]\) | \(2654208\) | \(3.2142\) | |
| 19110.bq3 | 19110bo8 | \([1, 1, 1, -121570716, 539085800763]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-10576819922457882813378750\) | \([2]\) | \(5308416\) | \(3.5607\) | |
| 19110.bq4 | 19110bo4 | \([1, 1, 1, -24323846, 46087901579]\) | \(13527956825588849127121/25701087819771000\) | \(3023707280908238379000\) | \([2]\) | \(1769472\) | \(3.0114\) | |
| 19110.bq5 | 19110bo3 | \([1, 1, 1, -7787326, 7996701803]\) | \(443915739051786565201/21894701746029840\) | \(2575889765718664646160\) | \([2]\) | \(1327104\) | \(2.8676\) | |
| 19110.bq6 | 19110bo2 | \([1, 1, 1, -2028846, 195873579]\) | \(7850236389974007121/4400862921000000\) | \(517757121792729000000\) | \([2, 2]\) | \(884736\) | \(2.6649\) | |
| 19110.bq7 | 19110bo1 | \([1, 1, 1, -1260526, -542328277]\) | \(1882742462388824401/11650189824000\) | \(1370633182603776000\) | \([2]\) | \(442368\) | \(2.3183\) | \(\Gamma_0(N)\)-optimal |
| 19110.bq8 | 19110bo5 | \([1, 1, 1, 7973034, 1564130763]\) | \(476437916651992691759/284661685546875000\) | \(-33490162642904296875000\) | \([2]\) | \(1769472\) | \(3.0114\) |
Rank
sage: E.rank()
The elliptic curves in class 19110.bq have rank \(1\).
Complex multiplication
The elliptic curves in class 19110.bq do not have complex multiplication.Modular form 19110.2.a.bq
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
