# Properties

 Label 722.e Number of curves $3$ Conductor $722$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 722.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
722.e1 722e3 $$[1, 1, 1, -30873, 16782247]$$ $$-69173457625/2550136832$$ $$-119973433931988992$$ $$[]$$ $$6480$$ $$1.9574$$
722.e2 722e1 $$[1, 1, 1, -5603, -163815]$$ $$-413493625/152$$ $$-7150973912$$ $$[]$$ $$720$$ $$0.85878$$ $$\Gamma_0(N)$$-optimal
722.e3 722e2 $$[1, 1, 1, 3422, -612177]$$ $$94196375/3511808$$ $$-165216101262848$$ $$[]$$ $$2160$$ $$1.4081$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form722.2.a.e

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 