# Properties

 Label 8450.x Number of curves $2$ Conductor $8450$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 8450.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8450.x1 8450r2 $$[1, 1, 1, -1914858, -1020687049]$$ $$-6434774386429585/140608$$ $$-16967198996800$$ $$[]$$ $$181440$$ $$2.0649$$
8450.x2 8450r1 $$[1, 1, 1, -22058, -1603529]$$ $$-9836106385/3407872$$ $$-411228681011200$$ $$[]$$ $$60480$$ $$1.5156$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 8450.x have rank $$0$$.

## Complex multiplication

The elliptic curves in class 8450.x do not have complex multiplication.

## Modular form8450.2.a.x

sage: E.q_eigenform(10)

$$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + 5 q^{7} + q^{8} + q^{9} + 3 q^{11} + 2 q^{12} + 5 q^{14} + q^{16} - 3 q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.