# Properties

 Label 8470.r Number of curves $4$ Conductor $8470$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.r1 8470u4 $$[1, 0, 0, -1893471, -774028199]$$ $$423783056881319689/99207416000000$$ $$175751989096376000000$$ $$[2]$$ $$414720$$ $$2.5965$$
8470.r2 8470u2 $$[1, 0, 0, -1770656, -907026680]$$ $$346553430870203929/8300600$$ $$14705019236600$$ $$[2]$$ $$138240$$ $$2.0472$$
8470.r3 8470u1 $$[1, 0, 0, -110536, -14214144]$$ $$-84309998289049/414124480$$ $$-733646777913280$$ $$[2]$$ $$69120$$ $$1.7006$$ $$\Gamma_0(N)$$-optimal
8470.r4 8470u3 $$[1, 0, 0, 274849, -75395495]$$ $$1296134247276791/2137096192000$$ $$-3785996266995712000$$ $$[2]$$ $$207360$$ $$2.2499$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.r

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.