Abelian variety isogeny class 2.3.ad_g over $\F_{3}$

• 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:	$1$
Isomorphism classes:	6

This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

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

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+2x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

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:

• 1.3.ad : \(\Q(\sqrt{-3}) \).
• 1.3.a : \(\Q(\sqrt{-3}) \).

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 $\infty$.

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.

Twist	Extension degree	Common 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