# Properties

 Label 8470.j Number of curves $4$ Conductor $8470$ CM no Rank $1$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.j1 8470h4 $$[1, -1, 0, -32390, 2251706]$$ $$2121328796049/120050$$ $$212675898050$$ $$$$ $$20480$$ $$1.2379$$
8470.j2 8470h3 $$[1, -1, 0, -10610, -390450]$$ $$74565301329/5468750$$ $$9688224218750$$ $$$$ $$20480$$ $$1.2379$$
8470.j3 8470h2 $$[1, -1, 0, -2140, 31356]$$ $$611960049/122500$$ $$217016222500$$ $$[2, 2]$$ $$10240$$ $$0.89136$$
8470.j4 8470h1 $$[1, -1, 0, 280, 2800]$$ $$1367631/2800$$ $$-4960370800$$ $$$$ $$5120$$ $$0.54479$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.j

sage: E.q_eigenform(10)

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