# Properties

 Label 8470.b Number of curves $2$ Conductor $8470$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 8470.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.b1 8470e1 $$[1, 0, 1, -179, -2994]$$ $$-5200020529/28672000$$ $$-3469312000$$ $$[]$$ $$6480$$ $$0.51463$$ $$\Gamma_0(N)$$-optimal
8470.b2 8470e2 $$[1, 0, 1, 1581, 73742]$$ $$3615170357711/21437500000$$ $$-2593937500000$$ $$[]$$ $$19440$$ $$1.0639$$

## Rank

sage: E.rank()

The elliptic curves in class 8470.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 8470.b do not have complex multiplication.

## Modular form8470.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - 2q^{12} - 2q^{13} + q^{14} + 2q^{15} + q^{16} - 3q^{17} - q^{18} + 7q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 