# Properties

 Label 338800w Number of curves 4 Conductor 338800 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("338800.w1")

sage: E.isogeny_class()

## Elliptic curves in class 338800w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
338800.w3 338800w1 [0, 1, 0, -2711408, 33440927188] [2] 39813120 $$\Gamma_0(N)$$-optimal
338800.w2 338800w2 [0, 1, 0, -173079408, 868584863188] [2] 79626240
338800.w4 338800w3 [0, 1, 0, 24392592, -900833952812] [2] 119439360
338800.w1 338800w4 [0, 1, 0, -1264015408, -16807519120812] [2] 238878720

## Rank

sage: E.rank()

The elliptic curves in class 338800w have rank $$1$$.

## Modular form 338800.2.a.w

sage: E.q_eigenform(10)

$$q - 2q^{3} - q^{7} + q^{9} - 4q^{13} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.