# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 690.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.e1 690e3 $$[1, 0, 1, -10644, 420826]$$ $$133345896593725369/340006815000$$ $$340006815000$$ $$$$ $$1920$$ $$1.0891$$
690.e2 690e2 $$[1, 0, 1, -924, 922]$$ $$87109155423289/49979073600$$ $$49979073600$$ $$[2, 2]$$ $$960$$ $$0.74257$$
690.e3 690e1 $$[1, 0, 1, -604, -5734]$$ $$24310870577209/114462720$$ $$114462720$$ $$$$ $$480$$ $$0.39599$$ $$\Gamma_0(N)$$-optimal
690.e4 690e4 $$[1, 0, 1, 3676, 8282]$$ $$5495662324535111/3207841648920$$ $$-3207841648920$$ $$$$ $$1920$$ $$1.0891$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## 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 LMFDB 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 LMFDB labels. 