# Properties

 Label 690j Number of curves $2$ Conductor $690$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 690j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.j2 690j1 $$[1, 0, 0, -245, -1503]$$ $$-1626794704081/8125440$$ $$-8125440$$ $$$$ $$240$$ $$0.17333$$ $$\Gamma_0(N)$$-optimal
690.j1 690j2 $$[1, 0, 0, -3925, -94975]$$ $$6687281588245201/165600$$ $$165600$$ $$$$ $$480$$ $$0.51991$$

## Rank

sage: E.rank()

The elliptic curves in class 690j have rank $$0$$.

## Complex multiplication

The elliptic curves in class 690j do not have complex multiplication.

## Modular form690.2.a.j

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - 2q^{11} + q^{12} + 4q^{13} + q^{15} + q^{16} + 6q^{17} + q^{18} - 8q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 