# Properties

 Label 2.81.abg_py Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 9 x )^{2}( 1 - 14 x + 81 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.216346895939$ Angle rank: $1$ (numerical) Jacobians: 3

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4352 41779200 282004486400 1852970355916800 12157651951711430912 79766320794946007040000 523347404273394977385128192 3433683554967536294019937075200 22528399313941250143604756860985600 147808829255717821926156950744494080000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 50 6366 530642 43045566 3486780530 282429103518 22876782455570 1853020045666686 150094633757983922 12157665446009349726

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.as $\times$ 1.81.ao and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.81.as : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.81.ao : $$\Q(\sqrt{-2})$$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.ae_adm $2$ (not in LMFDB) 2.81.e_adm $2$ (not in LMFDB) 2.81.bg_py $2$ (not in LMFDB) 2.81.af_bk $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.ae_adm $2$ (not in LMFDB) 2.81.e_adm $2$ (not in LMFDB) 2.81.bg_py $2$ (not in LMFDB) 2.81.af_bk $3$ (not in LMFDB) 2.81.ao_gg $4$ (not in LMFDB) 2.81.o_gg $4$ (not in LMFDB) 2.81.ax_lc $6$ (not in LMFDB) 2.81.f_bk $6$ (not in LMFDB) 2.81.x_lc $6$ (not in LMFDB)