# Properties

 Label 222.e Number of curves $2$ Conductor $222$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 222.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
222.e1 222a2 $$[1, 0, 0, -148, -706]$$ $$-358667682625/303918$$ $$-303918$$ $$[]$$ $$36$$ $$-0.020627$$
222.e2 222a1 $$[1, 0, 0, 2, -4]$$ $$857375/7992$$ $$-7992$$ $$$$ $$12$$ $$-0.56993$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 222.e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 222.e do not have complex multiplication.

## Modular form222.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} + 3 q^{11} + q^{12} - q^{13} - q^{14} + q^{16} - 3 q^{17} + q^{18} - 7 q^{19} + O(q^{20})$$

## 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. 