# Properties

 Label 429429y Number of curves 6 Conductor 429429 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("429429.y1")

sage: E.isogeny_class()

## Elliptic curves in class 429429y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
429429.y4 429429y1 [1, 1, 1, -695692, -223561876] [2] 4608000 $$\Gamma_0(N)$$-optimal*
429429.y3 429429y2 [1, 1, 1, -797937, -153667194] [2, 2] 9216000 $$\Gamma_0(N)$$-optimal*
429429.y2 429429y3 [1, 1, 1, -5807942, 5277178226] [2, 2] 18432000 $$\Gamma_0(N)$$-optimal*
429429.y6 429429y4 [1, 1, 1, 2576148, -1109208066] [2] 18432000
429429.y1 429429y5 [1, 1, 1, -92409457, 341879946728] [2] 36864000 $$\Gamma_0(N)$$-optimal*
429429.y5 429429y6 [1, 1, 1, 633493, 16348716704] [2] 36864000
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 429429y1.

## Rank

sage: E.rank()

The elliptic curves in class 429429y have rank $$1$$.

## Modular form 429429.2.a.y

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.