# Properties

 Label 570.b Number of curves $2$ Conductor $570$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 570.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
570.b1 570a2 $$[1, 1, 0, -1618, 24388]$$ $$468898230633769/5540400$$ $$5540400$$ $$$$ $$384$$ $$0.44399$$
570.b2 570a1 $$[1, 1, 0, -98, 372]$$ $$-105756712489/12476160$$ $$-12476160$$ $$$$ $$192$$ $$0.097417$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 570.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 570.b do not have complex multiplication.

## Modular form570.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} - 2 q^{14} + q^{15} + q^{16} + 2 q^{17} - q^{18} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 