# Properties

 Label 199410.bo Number of curves $6$ Conductor $199410$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("199410.bo1")

sage: E.isogeny_class()

## Elliptic curves in class 199410.bo

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
199410.bo1 199410y3 [1, 1, 1, -31905606, 69353018919] [2] 7077888
199410.bo2 199410y6 [1, 1, 1, -7464876, -6727521177] [2] 14155776
199410.bo3 199410y4 [1, 1, 1, -2046126, 1023458823] [2, 2] 7077888
199410.bo4 199410y2 [1, 1, 1, -1994106, 1083011319] [2, 2] 3538944
199410.bo5 199410y1 [1, 1, 1, -121386, 17808183] [2] 1769472 $$\Gamma_0(N)$$-optimal
199410.bo6 199410y5 [1, 1, 1, 2540304, 4964119479] [2] 14155776

## Rank

sage: E.rank()

The elliptic curves in class 199410.bo have rank $$1$$.

## Modular form 199410.2.a.bo

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 2q^{13} + q^{15} + q^{16} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.