# Properties

 Label 441090.bw Number of curves $4$ Conductor $441090$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bw1")

sage: E.isogeny_class()

## Elliptic curves in class 441090.bw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441090.bw1 441090bw3 $$[1, -1, 0, -12704184, 15960215740]$$ $$64443098670429961/6032611833300$$ $$21227215250959147041300$$ $$[2]$$ $$66060288$$ $$3.0225$$ $$\Gamma_0(N)$$-optimal*
441090.bw2 441090bw2 $$[1, -1, 0, -2817684, -1540866560]$$ $$703093388853961/115124490000$$ $$405093580925806890000$$ $$[2, 2]$$ $$33030144$$ $$2.6759$$ $$\Gamma_0(N)$$-optimal*
441090.bw3 441090bw1 $$[1, -1, 0, -2696004, -1703114672]$$ $$615882348586441/21715200$$ $$76410224518867200$$ $$[2]$$ $$16515072$$ $$2.3294$$ $$\Gamma_0(N)$$-optimal*
441090.bw4 441090bw4 $$[1, -1, 0, 5121936, -8659529852]$$ $$4223169036960119/11647532812500$$ $$-40984683415027157812500$$ $$[2]$$ $$66060288$$ $$3.0225$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 441090.bw1.

## Rank

sage: E.rank()

The elliptic curves in class 441090.bw have rank $$1$$.

## Complex multiplication

The elliptic curves in class 441090.bw do not have complex multiplication.

## Modular form 441090.2.a.bw

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} + 4 q^{7} - q^{8} - q^{10} - 4 q^{11} - 4 q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.