# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 19110.bq

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

The elliptic curves in class 19110.bq 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 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.