# Properties

 Label 722.c Number of curves $2$ Conductor $722$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 722.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
722.c1 722a2 $$[1, 0, 1, -33581, -2375576]$$ $$-246579625/512$$ $$-8695584276992$$ $$[]$$ $$2052$$ $$1.3686$$
722.c2 722a1 $$[1, 0, 1, 714, -16080]$$ $$2375/8$$ $$-135868504328$$ $$$$ $$684$$ $$0.81932$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 722.c have rank $$1$$.

## Complex multiplication

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

## Modular form722.2.a.c

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 