Abelian variety isogeny class 5.2.ag_u_abx_dq_afq over $\F_{2}$

• Label: 5.2.ag_u_abx_dq_afq
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 6 x + 20 x^{2} - 49 x^{3} + 94 x^{4} - 146 x^{5} + 188 x^{6} - 196 x^{7} + 160 x^{8} - 96 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$, $\pm0.583333333333$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	4

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$1976$	$26600$	$462384$	$27928582$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$9$	$6$	$1$	$27$	$78$	$165$	$337$	$582$	$1029$

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 $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ab $\times$ 2.2.ad_f $\times$ 2.2.ac_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 2.2.ad_f : \(\Q(\sqrt{-3}, \sqrt{5})\).
• 2.2.ac_c : \(\Q(\zeta_{12})\).

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h^2 $\times$ 1.4096.bv $\times$ 1.4096.ey^2. The endomorphism algebra for each factor is:

• 1.4096.h^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
• 1.4096.bv : \(\Q(\sqrt{-7}) \).
• 1.4096.ey^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 2.4.a_ae $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:

  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.a_ae : \(\Q(\zeta_{12})\).
  • 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae^2 $\times$ 1.8.f $\times$ 2.8.a_l. The endomorphism algebra for each factor is:

  • 1.8.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.8.f : \(\Q(\sqrt{-7}) \).
  • 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae^2 $\times$ 1.16.ab $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is:

  • 1.16.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.16.ab : \(\Q(\sqrt{-7}) \).
  • 2.16.ah_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.a^2 $\times$ 1.64.l^2. The endomorphism algebra for each factor is:

  • 1.64.aj : \(\Q(\sqrt{-7}) \).
  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.64.l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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
5.2.ae_k_ax_bq_ack	$2$	(not in LMFDB)
5.2.ac_e_ab_ag_o	$2$	(not in LMFDB)
5.2.a_c_ab_ac_ac	$2$	(not in LMFDB)
5.2.a_c_b_ac_c	$2$	(not in LMFDB)
5.2.c_e_b_ag_ao	$2$	(not in LMFDB)
5.2.e_k_x_bq_ck	$2$	(not in LMFDB)
5.2.g_u_bx_dq_fq	$2$	(not in LMFDB)
5.2.ad_f_ah_k_ao	$3$	(not in LMFDB)
5.2.a_c_ab_e_ai	$3$	(not in LMFDB)
5.2.d_f_f_k_q	$3$	(not in LMFDB)
5.2.g_u_bv_dk_fg	$3$	(not in LMFDB)