# Properties

 Label 690e Number of curves $4$ Conductor $690$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("690.e1")

sage: E.isogeny_class()

## Elliptic curves in class 690e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.e3 690e1 [1, 0, 1, -604, -5734]  480 $$\Gamma_0(N)$$-optimal
690.e2 690e2 [1, 0, 1, -924, 922] [2, 2] 960
690.e1 690e3 [1, 0, 1, -10644, 420826]  1920
690.e4 690e4 [1, 0, 1, 3676, 8282]  1920

## Rank

sage: E.rank()

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

## Modular form690.2.a.e

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 