# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("86640.j1")

sage: E.isogeny_class()

## Elliptic curves in class 86640bw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86640.j3 86640bw1 [0, -1, 0, -179176, 29077360]  552960 $$\Gamma_0(N)$$-optimal
86640.j2 86640bw2 [0, -1, 0, -294696, -12879504] [2, 2] 1105920
86640.j4 86640bw3 [0, -1, 0, 1149304, -102985104]  2211840
86640.j1 86640bw4 [0, -1, 0, -3587016, -2609861520]  2211840

## Rank

sage: E.rank()

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

## Modular form 86640.2.a.j

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. 