# Properties

 Label 16731.k Number of curves 4 Conductor 16731 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("16731.k1")
sage: E.isogeny_class()

## Elliptic curves in class 16731.k

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
16731.k1 16731k3 [1, -1, 0, -222858, -40398539] 2 110592
16731.k2 16731k2 [1, -1, 0, -17523, -276080] 4 55296
16731.k3 16731k1 [1, -1, 0, -9918, 379471] 2 27648 $$\Gamma_0(N)$$-optimal
16731.k4 16731k4 [1, -1, 0, 66132, -2200145] 2 110592

## Rank

sage: E.rank()

The elliptic curves in class 16731.k have rank $$1$$.

## Modular form 16731.2.a.k

sage: E.q_eigenform(10)
$$q + q^{2} - q^{4} - 2q^{5} - 4q^{7} - 3q^{8} - 2q^{10} + q^{11} - 4q^{14} - q^{16} + 2q^{17} + O(q^{20})$$

## 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. 