# Properties

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

# Related objects

## 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, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

## 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 (of which all are hyperelliptic), and hence is principally polarizable:

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $80$ $436$ $6400$ $69044$

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

## 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.
TwistExtension degreeCommon 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)