# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("333270.bs1")

sage: E.isogeny_class()

## Elliptic curves in class 333270.bs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.bs1 333270bs3 [1, -1, 0, -222279, 35401085] [2] 3649536
333270.bs2 333270bs1 [1, -1, 0, -55644, -5031972] [2] 1216512 $$\Gamma_0(N)$$-optimal
333270.bs3 333270bs2 [1, -1, 0, -39774, -7974270] [2] 2433024
333270.bs4 333270bs4 [1, -1, 0, 349041, 187029413] [2] 7299072

## Rank

sage: E.rank()

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

## Modular form 333270.2.a.bs

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.