Properties

 Label 2.3.ac_h Base field $\F_{3}$ Dimension $2$ $p$-rank $2$ Ordinary yes Supersingular no Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes

Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - x + 3 x^{2} )^{2}$ $1 - 2 x + 7 x^{2} - 6 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.406785250661$, $\pm0.406785250661$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary. $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+x^4+2x^3+x^2+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9$ $225$ $1296$ $5625$ $45369$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $20$ $44$ $68$ $182$ $710$ $2354$ $6788$ $19412$ $58100$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_f$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.b_ac$3$2.27.q_eo
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_f$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.b_ac$3$2.27.q_eo
2.3.a_af$4$2.81.ao_id
2.3.ab_ac$6$2.729.au_chy