# Properties

 Label 2.3.ac_c Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.116139763599$, $\pm0.616139763599$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{5})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not 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 2 curves, and hence is principally polarizable:

• $y^2=2x^5+x^4+x^3+x^2+x+2$
• $y^2=2x^6+2x^5+x^4+x^2+x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 80 436 6400 69044 531920 4840196 45158400 392133604 3486722000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 10 14 78 282 730 2214 6878 19922 59050

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 2.9.a_ac and its endomorphism algebra is $$\Q(i, \sqrt{5})$$.

## 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 2.3.c_c $2$ 2.9.a_ac 2.3.a_ae $8$ (not in LMFDB) 2.3.a_e $8$ (not in LMFDB)