# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 8470.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.m1 8470i1 $$[1, 0, 1, -4964, 227962]$$ $$-63088729/68600$$ $$-14705019236600$$ $$$$ $$19008$$ $$1.2214$$ $$\Gamma_0(N)$$-optimal
8470.m2 8470i2 $$[1, 0, 1, 41621, -4132394]$$ $$37199299511/56000000$$ $$-12004097336000000$$ $$[]$$ $$57024$$ $$1.7707$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.m

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 