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

• Label: 5.2.ah_z_acj_el_agw
• Base field: $\F_{2}$
• 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:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 50 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8} )$
	$1 - 7 x + 25 x^{2} - 61 x^{3} + 115 x^{4} - 178 x^{5} + 230 x^{6} - 244 x^{7} + 200 x^{8} - 112 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.00978468837242$, $\pm0.190215311628$, $\pm0.250000000000$, $\pm0.409784688372$, $\pm0.609784688372$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$1205$	$24583$	$1090525$	$38101136$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$6$	$8$	$18$	$41$	$54$	$108$	$242$	$386$	$781$

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 $\F_{2^{20}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac $\times$ 4.2.af_n_az_bn 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.ac : \(\Q(\sqrt{-1}) \).
• 4.2.af_n_az_bn : \(\Q(\zeta_{15})\).

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

The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.acmj^4 $\times$ 1.1048576.dau. The endomorphism algebra for each factor is:

• 1.1048576.acmj^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$
• 1.1048576.dau : 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.a $\times$ 4.4.b_ad_ah_f. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 4.4.b_ad_ah_f : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 4.16.ah_bh_aep_lt. The endomorphism algebra for each factor is:

  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 4.16.ah_bh_aep_lt : \(\Q(\zeta_{15})\).
• Endomorphism algebra over $\F_{2^{5}}$
  The base change of $A$ to $\F_{2^{5}}$ is 1.32.i $\times$ 2.32.a_acj^2. The endomorphism algebra for each factor is:

  • 1.32.i : \(\Q(\sqrt{-1}) \).
  • 2.32.a_acj^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{5})\)$)$
• Endomorphism algebra over $\F_{2^{10}}$
  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acj^4 $\times$ 1.1024.a. The endomorphism algebra for each factor is:

  • 1.1024.acj^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-15}) \)$)$
  • 1.1024.a : \(\Q(\sqrt{-1}) \).

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
5.2.ad_f_aj_p_aw	$2$	(not in LMFDB)
5.2.d_f_j_p_w	$2$	(not in LMFDB)
5.2.h_z_cj_el_gw	$2$	(not in LMFDB)
5.2.ab_e_ae_n_ak	$3$	(not in LMFDB)
5.2.c_ac_ah_b_o	$3$	(not in LMFDB)