Abelian variety isogeny class 5.3.ak_by_agj_po_abea over $\F_{3}$

• Label: 5.3.ak_by_agj_po_abea
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $4$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$
	$1 - 10 x + 50 x^{2} - 165 x^{3} + 404 x^{4} - 780 x^{5} + 1212 x^{6} - 1485 x^{7} + 1350 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.304086723985$, $\pm0.560185743604$
Angle rank:	$4$ (numerical)

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

Newton polygon

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10$	$65100$	$17401720$	$4231500000$	$1047034665050$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$10$	$33$	$98$	$294$	$727$	$2003$	$6498$	$20229$	$59750$

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 $\times$ 1.3.ac $\times$ 3.3.af_n_az 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.ac : \(\Q(\sqrt{-2}) \).
• 3.3.af_n_az : 6.0.1342367.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc $\times$ 3.729.al_cmn_aoxb. The endomorphism algebra for each factor is:

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 3.729.al_cmn_aoxb : 6.0.1342367.1.

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.c $\times$ 3.9.b_ad_ah. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 3.9.b_ad_ah : 6.0.1342367.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k $\times$ 3.27.af_h_ev. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 1.27.k : \(\Q(\sqrt{-2}) \).
  • 3.27.af_h_ev : 6.0.1342367.1.

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
5.3.ag_s_abp_dk_agm	$2$	(not in LMFDB)
5.3.ae_i_aj_i_am	$2$	(not in LMFDB)
5.3.a_a_af_e_a	$2$	(not in LMFDB)
5.3.a_a_f_e_a	$2$	(not in LMFDB)
5.3.e_i_j_i_m	$2$	(not in LMFDB)
5.3.g_s_bp_dk_gm	$2$	(not in LMFDB)
5.3.k_by_gj_po_bea	$2$	(not in LMFDB)
5.3.ah_bd_adj_hy_apg	$3$	(not in LMFDB)
5.3.ae_i_aj_i_am	$3$	(not in LMFDB)