# Properties

 Label 722.b Number of curves $2$ Conductor $722$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 722.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
722.b1 722c2 $$[1, 0, 1, -25278, 1710222]$$ $$-37966934881/4952198$$ $$-232980517796438$$ $$[]$$ $$3600$$ $$1.4900$$
722.b2 722c1 $$[1, 0, 1, -8, -8138]$$ $$-1/608$$ $$-28603895648$$ $$[]$$ $$720$$ $$0.68529$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 722.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 722.b do not have complex multiplication.

## Modular form722.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} - q^{8} - 2 q^{9} + 4 q^{10} + 2 q^{11} + q^{12} + q^{13} - 3 q^{14} - 4 q^{15} + q^{16} + 3 q^{17} + 2 q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 