# Properties

 Label 8470.y Number of curves $2$ Conductor $8470$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("y1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.y1 8470w2 $$[1, -1, 1, -99243, -12008719]$$ $$45844273539/350$$ $$825281691850$$ $$$$ $$42240$$ $$1.4612$$
8470.y2 8470w1 $$[1, -1, 1, -6073, -194763]$$ $$-10503459/980$$ $$-2310788737180$$ $$$$ $$21120$$ $$1.1146$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.y

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 