# Properties

 Label 690k Number of curves 6 Conductor 690 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("690.k1")

sage: E.isogeny_class()

## Elliptic curves in class 690k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
690.k5 690k1 [1, 0, 0, -420, 3600]  384 $$\Gamma_0(N)$$-optimal
690.k4 690k2 [1, 0, 0, -6900, 220032] [2, 4] 768
690.k3 690k3 [1, 0, 0, -7080, 207900] [2, 2] 1536
690.k1 690k4 [1, 0, 0, -110400, 14109732]  1536
690.k2 690k5 [1, 0, 0, -25830, -1370850]  3072
690.k6 690k6 [1, 0, 0, 8790, 1010922]  3072

## Rank

sage: E.rank()

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

## Modular form690.2.a.k

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 