# Properties

 Label 8470.c Number of curves $4$ Conductor $8470$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 8470.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.c1 8470f4 $$[1, 0, 1, -3160039, 2100623886]$$ $$1969902499564819009/63690429687500$$ $$112831481307617187500$$ $$[2]$$ $$414720$$ $$2.6216$$
8470.c2 8470f2 $$[1, 0, 1, -432699, -108616378]$$ $$5057359576472449/51765560000$$ $$91705847239160000$$ $$[2]$$ $$138240$$ $$2.0723$$
8470.c3 8470f1 $$[1, 0, 1, -6779, -4180794]$$ $$-19443408769/4249907200$$ $$-7528969849139200$$ $$[2]$$ $$69120$$ $$1.7257$$ $$\Gamma_0(N)$$-optimal
8470.c4 8470f3 $$[1, 0, 1, 60981, 112610342]$$ $$14156681599871/3100231750000$$ $$-5492249659261750000$$ $$[2]$$ $$207360$$ $$2.2750$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form8470.2.a.c

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.