Properties

Label 19110bo
Number of curves 8
Conductor 19110
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("19110.bq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 19110bo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19110.bq7 19110bo1 [1, 1, 1, -1260526, -542328277] [2] 442368 \(\Gamma_0(N)\)-optimal
19110.bq6 19110bo2 [1, 1, 1, -2028846, 195873579] [2, 2] 884736  
19110.bq5 19110bo3 [1, 1, 1, -7787326, 7996701803] [2] 1327104  
19110.bq4 19110bo4 [1, 1, 1, -24323846, 46087901579] [2] 1769472  
19110.bq8 19110bo5 [1, 1, 1, 7973034, 1564130763] [2] 1769472  
19110.bq2 19110bo6 [1, 1, 1, -123083346, 525537476379] [2, 2] 2654208  
19110.bq1 19110bo7 [1, 1, 1, -1969332296, 33636904645259] [2] 5308416  
19110.bq3 19110bo8 [1, 1, 1, -121570716, 539085800763] [2] 5308416  

Rank

sage: E.rank()
 

The elliptic curves in class 19110bo have rank \(1\).

Modular form 19110.2.a.bq

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{12} - q^{13} + q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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.