# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 + 3 x^{2} )$ $1 - 3 x + 6 x^{2} - 9 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.5$ Angle rank: $0$ (numerical) Jacobians: 1

This isogeny class is not 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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+2x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $112$ $784$ $5824$ $66124$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $13$ $28$ $73$ $271$ $838$ $2269$ $6481$ $19684$ $59293$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 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 $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.ad : $$\Q(\sqrt{-3})$$. 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$

## 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.ag_p$3$2.27.a_cc
2.3.a_ad$3$2.27.a_cc
2.3.a_g$3$2.27.a_cc
2.3.d_g$3$2.27.a_cc
2.3.g_p$3$2.27.a_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ag_p$3$2.27.a_cc
2.3.a_ad$3$2.27.a_cc
2.3.a_g$3$2.27.a_cc
2.3.d_g$3$2.27.a_cc
2.3.g_p$3$2.27.a_cc
2.3.a_ad$6$2.729.ee_gmg
2.3.a_ag$12$(not in LMFDB)
2.3.a_d$12$(not in LMFDB)
2.3.a_a$24$(not in LMFDB)