# Properties

 Label 333270de Number of curves $4$ Conductor $333270$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("333270.de1")

sage: E.isogeny_class()

## Elliptic curves in class 333270de

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.de3 333270de1 [1, -1, 1, -24698, -1302919] [2] 1216512 $$\Gamma_0(N)$$-optimal
333270.de4 333270de2 [1, -1, 1, 38782, -6939943] [2] 2433024
333270.de1 333270de3 [1, -1, 1, -500798, 136364041] [2] 3649536
333270.de2 333270de4 [1, -1, 1, -357968, 215663257] [2] 7299072

## Rank

sage: E.rank()

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

## Modular form 333270.2.a.de

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} + 2q^{13} - q^{14} + q^{16} - 2q^{19} + O(q^{20})$$

## 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.