# Properties

 Label 266805bn Number of curves $6$ Conductor $266805$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bn1")

sage: E.isogeny_class()

## Elliptic curves in class 266805bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
266805.bn6 266805bn1 [1, -1, 1, 1866523, 285120242124] [2] 44236800 $$\Gamma_0(N)$$-optimal
266805.bn5 266805bn2 [1, -1, 1, -638732282, 6103294828656] [2, 2] 88473600
266805.bn2 266805bn3 [1, -1, 1, -10169273687, 394716838942374] [2, 2] 176947200
266805.bn4 266805bn4 [1, -1, 1, -1357771757, -10154762932674] [2] 176947200
266805.bn1 266805bn5 [1, -1, 1, -162708362312, 25261761097860774] [2] 353894400
266805.bn3 266805bn6 [1, -1, 1, -10118847542, 398825036805066] [2] 353894400

## Rank

sage: E.rank()

The elliptic curves in class 266805bn have rank $$1$$.

## Complex multiplication

The elliptic curves in class 266805bn do not have complex multiplication.

## Modular form 266805.2.a.bn

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} - 2q^{13} - q^{16} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.