Abelian variety isogeny class 4.3.am_co_aii_rr over $\F_{3}$

• Label: 4.3.am_co_aii_rr
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-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:	$4$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )^{4}$
	$1 - 12 x + 66 x^{2} - 216 x^{3} + 459 x^{4} - 648 x^{5} + 594 x^{6} - 324 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$
Angle rank:	$0$ (numerical)

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:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$2401$	$614656$	$68574961$	$5393580481$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-8$	$-2$	$28$	$118$	$352$	$946$	$2512$	$6886$	$19684$	$58078$

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^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, 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^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

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

  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

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
4.3.aj_bn_aee_ii	$3$	(not in LMFDB)
4.3.ag_m_a_abb	$3$	(not in LMFDB)
4.3.ag_v_acc_ee	$3$	(not in LMFDB)
4.3.ad_d_a_a	$3$	(not in LMFDB)
4.3.ad_m_abb_cc	$3$	(not in LMFDB)
4.3.a_ag_a_bb	$3$	(not in LMFDB)
4.3.a_d_a_a	$3$	(not in LMFDB)
4.3.a_m_a_cc	$3$	(not in LMFDB)
4.3.d_d_a_a	$3$	(not in LMFDB)
4.3.d_m_bb_cc	$3$	(not in LMFDB)
4.3.g_m_a_abb	$3$	(not in LMFDB)
4.3.g_v_cc_ee	$3$	(not in LMFDB)
4.3.j_bn_ee_ii	$3$	(not in LMFDB)
4.3.m_co_ii_rr	$3$	(not in LMFDB)