# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("690.a1")

sage: E.isogeny_class()

## Elliptic curves in class 690.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.a1 690a2 [1, 1, 0, -1748, -25392]  896
690.a2 690a1 [1, 1, 0, 172, -1968]  448 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form690.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + q^{10} + 6q^{11} - q^{12} - 2q^{13} + 2q^{14} + q^{15} + q^{16} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 LMFDB 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 LMFDB labels. 