# Properties

 Label 333270.cy Number of curves $2$ Conductor $333270$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 333270.cy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.cy1 333270cy1 $$[1, -1, 1, -2606172293, 21411682137981]$$ $$489781415227546051766883/233890092903563264000$$ $$934851451444332634457505792000$$ $$[2]$$ $$544997376$$ $$4.4447$$ $$\Gamma_0(N)$$-optimal
333270.cy2 333270cy2 $$[1, -1, 1, 9354475387, 162848733083517]$$ $$22649115256119592694355357/15973509811739648000000$$ $$-63845623586639738562130944000000$$ $$[2]$$ $$1089994752$$ $$4.7913$$

## Rank

sage: E.rank()

The elliptic curves in class 333270.cy have rank $$1$$.

## Complex multiplication

The elliptic curves in class 333270.cy do not have complex multiplication.

## Modular form 333270.2.a.cy

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.