# Properties

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

# Learn more

Show commands for: SageMath
sage: E = EllipticCurve("bp1")

sage: E.isogeny_class()

## Elliptic curves in class 333270bp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.bp1 333270bp1 $$[1, -1, 0, -674574, 209365380]$$ $$8493409990827/185150000$$ $$740038810905450000$$ $$[2]$$ $$5406720$$ $$2.2171$$ $$\Gamma_0(N)$$-optimal
333270.bp2 333270bp2 $$[1, -1, 0, 55446, 638179128]$$ $$4716275733/44023437500$$ $$-175960315093007812500$$ $$[2]$$ $$10813440$$ $$2.5636$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 333270.2.a.bp

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} - 2q^{11} + 2q^{13} + q^{14} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.