Properties

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

Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 + x - x^{2} + 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.281846319079$, $\pm0.873133061559$ Angle rank: $2$ (numerical) Number field: 4.0.10933.1 Galois group: $D_{4}$ Jacobians: 1

This isogeny class is simple and geometrically 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:

• $y^2=2x^5+2x^4+2x^3+2x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $13$ $65$ $1183$ $8125$ $58448$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $7$ $41$ $99$ $240$ $739$ $2007$ $6611$ $19463$ $59782$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 4.0.10933.1.
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.
TwistExtension degreeCommon base change
2.3.ab_ab$2$2.9.ad_n