# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("86640.di1")

sage: E.isogeny_class()

## Elliptic curves in class 86640.di

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86640.di1 86640eh4 [0, 1, 0, -17559160, 28314830420] [2] 3317760
86640.di2 86640eh3 [0, 1, 0, -1270840, 292959188] [2] 3317760
86640.di3 86640eh2 [0, 1, 0, -1097560, 442049300] [2, 2] 1658880
86640.di4 86640eh1 [0, 1, 0, -57880, 9126548] [2] 829440 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 86640.2.a.di

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - 4q^{7} + q^{9} + 4q^{11} + 2q^{13} + q^{15} - 2q^{17} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.