Properties

 Label 3.3.ae_g_ah Base Field $\F_{3}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $1 - 4 x + 6 x^{2} - 7 x^{3} + 18 x^{4} - 36 x^{5} + 27 x^{6}$ Frobenius angles: $\pm0.0452398905210$, $\pm0.239335307006$, $\pm0.691360448188$ Angle rank: $3$ (numerical) Number field: 6.0.2461019.1 Galois group: $D_{6}$ Jacobians: 1

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $x^4+2x^3z+2x^2y^2+x^2yz+x^2z^2+xy^2z+y^4+z^4=0$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 5 495 11465 636075 14736775 350725815 10376297545 272317065075 7417764013760 206718770265975

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 6 15 98 250 657 2170 6322 19140 59286

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 6.0.2461019.1.
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.e_g_h $2$ 3.9.ae_q_acp