# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 690b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.b2 690b1 $$[1, 1, 0, 167, -347]$$ $$510273943271/370215360$$ $$-370215360$$ $$$$ $$336$$ $$0.33248$$ $$\Gamma_0(N)$$-optimal
690.b1 690b2 $$[1, 1, 0, -753, -3843]$$ $$47316161414809/22001657400$$ $$22001657400$$ $$$$ $$672$$ $$0.67905$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form690.2.a.b

sage: E.q_eigenform(10)

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