Abelian variety isogeny class 4.5.al_cg_ahp_te over $\F_{5}$

• Label: 4.5.al_cg_ahp_te
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 8 x^{2} - 15 x^{3} + 25 x^{4} )$
	$1 - 11 x + 58 x^{2} - 197 x^{3} + 498 x^{4} - 985 x^{5} + 1450 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.206741677780$, $\pm0.540075011113$
Angle rank:	$2$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$332800$	$235286272$	$159488409600$	$105937056318784$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$21$	$118$	$653$	$3455$	$16458$	$78899$	$391485$	$1955998$	$9762381$

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 $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae^2 $\times$ 2.5.ad_i and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 2.5.ad_i : \(\Q(\sqrt{-3}, \sqrt{17})\).

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ha^2 $\times$ 1.15625.ja^2. The endomorphism algebra for each factor is:

• 1.15625.ha^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
• 1.15625.ja^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$
  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag^2 $\times$ 2.25.h_y. The endomorphism algebra for each factor is:

  • 1.25.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.25.h_y : \(\Q(\sqrt{-3}, \sqrt{17})\).
• Endomorphism algebra over $\F_{5^{3}}$
  The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae^2 $\times$ 2.125.a_ha. The endomorphism algebra for each factor is:

  • 1.125.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.125.a_ha : \(\Q(\sqrt{-3}, \sqrt{17})\).

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
4.5.ad_c_d_c	$2$	(not in LMFDB)
4.5.d_c_ad_c	$2$	(not in LMFDB)
4.5.f_k_l_s	$2$	(not in LMFDB)
4.5.l_cg_hp_te	$2$	(not in LMFDB)
4.5.ai_t_q_afc	$3$	(not in LMFDB)
4.5.af_k_al_s	$3$	(not in LMFDB)
4.5.b_h_e_s	$3$	(not in LMFDB)
4.5.e_e_ai_abb	$3$	(not in LMFDB)
4.5.h_bf_dw_jy	$3$	(not in LMFDB)