# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.o1 8470n1 $$[1, 0, 1, -16943, 2456056]$$ $$-2509090441/10718750$$ $$-2297659255718750$$ $$$$ $$57024$$ $$1.6326$$ $$\Gamma_0(N)$$-optimal
8470.o2 8470n2 $$[1, 0, 1, 149432, -59235794]$$ $$1721540467559/8070721400$$ $$-1730030808166753400$$ $$[]$$ $$171072$$ $$2.1819$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.o

sage: E.q_eigenform(10)

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