# Properties

 Label 16731.c Number of curves $6$ Conductor $16731$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("16731.c1")

sage: E.isogeny_class()

## Elliptic curves in class 16731.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
16731.c1 16731h3 [1, -1, 1, -10440176, -12981429340] [2] 344064
16731.c2 16731h5 [1, -1, 1, -4576721, 3650404502] [2] 688128
16731.c3 16731h4 [1, -1, 1, -720986, -157519384] [2, 2] 344064
16731.c4 16731h2 [1, -1, 1, -652541, -202693084] [2, 2] 172032
16731.c5 16731h1 [1, -1, 1, -36536, -3846670] [2] 86016 $$\Gamma_0(N)$$-optimal
16731.c6 16731h6 [1, -1, 1, 2039629, -1075147810] [2] 688128

## Rank

sage: E.rank()

The elliptic curves in class 16731.c have rank $$0$$.

## Modular form 16731.2.a.c

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.