# Properties

 Label 8470.z Number of curves $4$ Conductor $8470$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.z1 8470ba4 $$[1, -1, 1, -2530012, 1549562831]$$ $$1010962818911303721/57392720$$ $$101674704435920$$ $$$$ $$122880$$ $$2.1543$$
8470.z2 8470ba3 $$[1, -1, 1, -264892, -12309041]$$ $$1160306142246441/634128110000$$ $$1123396628679710000$$ $$$$ $$122880$$ $$2.1543$$
8470.z3 8470ba2 $$[1, -1, 1, -158412, 24149711]$$ $$248158561089321/1859334400$$ $$3293924308998400$$ $$[2, 2]$$ $$61440$$ $$1.8078$$
8470.z4 8470ba1 $$[1, -1, 1, -3532, 855759]$$ $$-2749884201/176619520$$ $$-312892253470720$$ $$$$ $$30720$$ $$1.4612$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.z

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 