# Properties

 Label 2.81.abg_pz Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $1 - 32 x + 415 x^{2} - 2592 x^{3} + 6561 x^{4}$ Frobenius angles: $\pm0.0549914688267$, $\pm0.208693592337$ Angle rank: $2$ (numerical) Number field: 4.0.166032.2 Galois group: $D_{4}$ Jacobians: 4

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 Jacobians of 4 curves, and hence is principally polarizable:

• $y^2=(2a^3+a^2)x^6+(a^3+2a^2+a)x^5+(a^3+2a^2+a)x^4+(a^3+2a+1)x^3+(2a^3+2a^2+1)x^2+2a^2x+a^2+a$
• $y^2=(a^2+2a)x^6+(a^3+1)x^5+(a^3+a^2+2a+2)x^4+(2a^3+2a^2+a+2)x^3+(2a^3+a+2)x^2+(2a^2+2a)x+2$
• $y^2=(a+2)x^6+x^5+(2a^2+a+2)x^4+(2a^2+a)x^3+(a^3+a^2+a+1)x^2+2a^3x+a^2+2a+2$
• $y^2=(a^3+a)x^6+(a^3+2a)x^5+(a^3+a^2+a+1)x^4+(a^2+2)x^3+(a^3+a^2+a+2)x^2+2a+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4353 41793153 282055751748 1853075345635977 12157805932625462433 79766501046260613602832 523347582479573769965280897 3433683709351342094908635746313 22528399435189910149350185590589892 147808829345946517727908170670779841473

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 50 6368 530738 43048004 3486824690 282429741734 22876790245394 1853020128981380 150094634565798674 12157665453430897568

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.166032.2.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

## 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 2.81.bg_pz $2$ (not in LMFDB)