# Properties

 Label 86640bw Number of curves $4$ Conductor $86640$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 86640bw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86640.j3 86640bw1 $$[0, -1, 0, -179176, 29077360]$$ $$3301293169/22800$$ $$4393558371532800$$ $$[2]$$ $$552960$$ $$1.8351$$ $$\Gamma_0(N)$$-optimal
86640.j2 86640bw2 $$[0, -1, 0, -294696, -12879504]$$ $$14688124849/8122500$$ $$1565205169858560000$$ $$[2, 2]$$ $$1105920$$ $$2.1817$$
86640.j4 86640bw3 $$[0, -1, 0, 1149304, -102985104]$$ $$871257511151/527800050$$ $$-101707031937409228800$$ $$[2]$$ $$2211840$$ $$2.5283$$
86640.j1 86640bw4 $$[0, -1, 0, -3587016, -2609861520]$$ $$26487576322129/44531250$$ $$8581168694400000000$$ $$[2]$$ $$2211840$$ $$2.5283$$

## Rank

sage: E.rank()

The elliptic curves in class 86640bw have rank $$1$$.

## Complex multiplication

The elliptic curves in class 86640bw do not have complex multiplication.

## Modular form 86640.2.a.bw

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} - 4q^{11} - 2q^{13} + q^{15} + 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.