# Properties

 Label 64009c Number of curves $2$ Conductor $64009$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 64009c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
64009.a2 64009c1 [1, 1, 1, -15881, 763720] [] 75504 $$\Gamma_0(N)$$-optimal
64009.a1 64009c2 [1, 1, 1, -161356, -97589018] [] 830544

## Rank

sage: E.rank()

The elliptic curves in class 64009c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 64009c do not have complex multiplication.

## Modular form 64009.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} + 2q^{3} - q^{4} - q^{5} - 2q^{6} - 2q^{7} + 3q^{8} + q^{9} + q^{10} - 2q^{12} - q^{13} + 2q^{14} - 2q^{15} - q^{16} - 5q^{17} - q^{18} + 6q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.