# Properties

 Label 8470.e Number of curves $4$ Conductor $8470$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.e1 8470l4 $$[1, 0, 1, -425923, 106370856]$$ $$4823468134087681/30382271150$$ $$53824046660765150$$ $$$$ $$138240$$ $$2.0478$$
8470.e2 8470l2 $$[1, 0, 1, -32673, -2178244]$$ $$2177286259681/105875000$$ $$187564020875000$$ $$$$ $$46080$$ $$1.4985$$
8470.e3 8470l3 $$[1, 0, 1, -10893, 3609428]$$ $$-80677568161/3131816380$$ $$-5548203757969180$$ $$$$ $$69120$$ $$1.7012$$
8470.e4 8470l1 $$[1, 0, 1, 1207, -131892]$$ $$109902239/4312000$$ $$-7638971032000$$ $$$$ $$23040$$ $$1.1519$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 8470.e have rank $$1$$.

## Complex multiplication

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

## Modular form8470.2.a.e

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 