# Properties

 Label 159936cy Number of curves 6 Conductor 159936 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("159936.z1")

sage: E.isogeny_class()

## Elliptic curves in class 159936cy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
159936.z5 159936cy1 [0, -1, 0, -106689, 12475233] [2] 1179648 $$\Gamma_0(N)$$-optimal
159936.z4 159936cy2 [0, -1, 0, -357569, -67756191] [2, 2] 2359296
159936.z6 159936cy3 [0, -1, 0, 708671, -395518367] [2] 4718592
159936.z2 159936cy4 [0, -1, 0, -5437889, -4878819231] [2, 2] 4718592
159936.z3 159936cy5 [0, -1, 0, -5155649, -5408132127] [2] 9437184
159936.z1 159936cy6 [0, -1, 0, -87005249, -312338826015] [2] 9437184

## Rank

sage: E.rank()

The elliptic curves in class 159936cy have rank $$1$$.

## Modular form 159936.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{5} + q^{9} - 4q^{11} - 2q^{13} + 2q^{15} - q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.