# Properties

 Label 690h Number of curves $2$ Conductor $690$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("690.g1")

sage: E.isogeny_class()

## Elliptic curves in class 690h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.g2 690h1 [1, 1, 1, 4, 29]  96 $$\Gamma_0(N)$$-optimal
690.g1 690h2 [1, 1, 1, -116, 413]  192

## Rank

sage: E.rank()

The elliptic curves in class 690h have rank $$1$$.

## Modular form690.2.a.g

sage: E.q_eigenform(10)

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

## 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. 