# Properties

 Label 2.3.a_a Base field $\F_{3}$ Dimension $2$ $p$-rank $0$ Ordinary no Supersingular yes Simple yes Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 + 9 x^{4}$ Frobenius angles: $\pm0.250000000000$, $\pm0.750000000000$ Angle rank: $0$ (numerical) Number field: $$\Q(i, \sqrt{6})$$ Galois group: $C_2^2$ Jacobians: 2

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2, 1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=x^5+2$
• $y^2=2x^5+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10$ $100$ $730$ $10000$ $59050$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $10$ $28$ $118$ $244$ $730$ $2188$ $6238$ $19684$ $59050$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{6})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{4}}$ is 1.81.s 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
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 1.9.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_ag$8$(not in LMFDB)
2.3.a_g$8$(not in LMFDB)
2.3.ag_p$24$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_ag$8$(not in LMFDB)
2.3.a_g$8$(not in LMFDB)
2.3.ag_p$24$(not in LMFDB)
2.3.ad_g$24$(not in LMFDB)
2.3.a_ad$24$(not in LMFDB)
2.3.a_d$24$(not in LMFDB)
2.3.d_g$24$(not in LMFDB)
2.3.g_p$24$(not in LMFDB)