# Properties

 Label 8470f Number of curves 4 Conductor 8470 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("8470.c1")

sage: E.isogeny_class()

## Elliptic curves in class 8470f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8470.c3 8470f1 [1, 0, 1, -6779, -4180794] [2] 69120 $$\Gamma_0(N)$$-optimal
8470.c2 8470f2 [1, 0, 1, -432699, -108616378] [2] 138240
8470.c4 8470f3 [1, 0, 1, 60981, 112610342] [2] 207360
8470.c1 8470f4 [1, 0, 1, -3160039, 2100623886] [2] 414720

## Rank

sage: E.rank()

The elliptic curves in class 8470f have rank $$0$$.

## Modular form8470.2.a.c

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.