# Properties

 Label 75810cy Number of curves $4$ Conductor $75810$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cy1")

sage: E.isogeny_class()

## Elliptic curves in class 75810cy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.cr3 75810cy1 [1, 0, 0, -1271, 10905] [2] 115200 $$\Gamma_0(N)$$-optimal
75810.cr2 75810cy2 [1, 0, 0, -8491, -293779] [2, 2] 230400
75810.cr4 75810cy3 [1, 0, 0, 2339, -989065] [2] 460800
75810.cr1 75810cy4 [1, 0, 0, -134841, -19069389] [2] 460800

## Rank

sage: E.rank()

The elliptic curves in class 75810cy have rank $$0$$.

## Complex multiplication

The elliptic curves in class 75810cy do not have complex multiplication.

## Modular form 75810.2.a.cy

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + O(q^{20})$$

## 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.