# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bt1")

sage: E.isogeny_class()

## Elliptic curves in class 333270bt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.bt1 333270bt1 $$[1, -1, 0, -23455550634, -578091962174860]$$ $$489781415227546051766883/233890092903563264000$$ $$681506708102918490519521722368000$$ $$[2]$$ $$1634992128$$ $$4.9940$$ $$\Gamma_0(N)$$-optimal
333270.bt2 333270bt2 $$[1, -1, 0, 84190278486, -4396999983533452]$$ $$22649115256119592694355357/15973509811739648000000$$ $$-46543459594660369411793458176000000$$ $$[2]$$ $$3269984256$$ $$5.3406$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 333270.2.a.bt

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 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.