Abelian variety isogeny class 3.3.aj_bk_add over $\F_{3}$

• Label: 3.3.aj_bk_add
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )^{3}$
	$1 - 9 x + 36 x^{2} - 81 x^{3} + 108 x^{4} - 81 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$
Angle rank:	$0$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$343$	$21952$	$753571$	$19902511$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$1$	$28$	$109$	$325$	$892$	$2431$	$6805$	$19684$	$58321$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, 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^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{3^{3}}$

  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

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
3.3.ad_a_j	$2$	3.9.aj_cc_ahh
3.3.d_a_aj	$2$	3.9.aj_cc_ahh
3.3.j_bk_dd	$2$	3.9.aj_cc_ahh
3.3.ag_s_abk	$3$	(not in LMFDB)
3.3.ad_a_j	$3$	(not in LMFDB)
3.3.ad_j_as	$3$	(not in LMFDB)
3.3.a_a_a	$3$	(not in LMFDB)
3.3.a_j_a	$3$	(not in LMFDB)
3.3.d_a_aj	$3$	(not in LMFDB)
3.3.d_j_s	$3$	(not in LMFDB)
3.3.g_s_bk	$3$	(not in LMFDB)
3.3.j_bk_dd	$3$	(not in LMFDB)