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

• Label: 4.5.am_cv_akw_bcy
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )^{2}$
	$1 - 12 x + 73 x^{2} - 282 x^{3} + 752 x^{4} - 1410 x^{5} + 1825 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.265942140215$, $\pm0.265942140215$, $\pm0.352416382350$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$72$	$466560$	$374409216$	$186624000000$	$98750488556232$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$28$	$180$	$752$	$3234$	$15478$	$77610$	$390432$	$1955124$	$9771628$

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 $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 1.5.ad^2 $\times$ 1.5.ac 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 : \(\Q(\sqrt{-1}) \).
• 1.5.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.5.ac : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{5^{4}}$ is 1.625.o^2 $\times$ 1.625.bx^2. The endomorphism algebra for each factor is:

• 1.625.o^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.625.bx^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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 $\times$ 1.25.b^2 $\times$ 1.25.g. The endomorphism algebra for each factor is:

  • 1.25.ag : \(\Q(\sqrt{-1}) \).
  • 1.25.b^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
  • 1.25.g : \(\Q(\sqrt{-1}) \).

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
4.5.ai_bh_adm_ia	$2$	(not in LMFDB)
4.5.ag_t_abk_cq	$2$	(not in LMFDB)
4.5.ae_j_g_abg	$2$	(not in LMFDB)
4.5.ac_d_am_ca	$2$	(not in LMFDB)
4.5.a_b_ag_bg	$2$	(not in LMFDB)
4.5.a_b_g_bg	$2$	(not in LMFDB)
4.5.c_d_m_ca	$2$	(not in LMFDB)
4.5.e_j_ag_abg	$2$	(not in LMFDB)
4.5.g_t_bk_cq	$2$	(not in LMFDB)
4.5.i_bh_dm_ia	$2$	(not in LMFDB)
4.5.m_cv_kw_bcy	$2$	(not in LMFDB)
4.5.ad_e_p_acg	$3$	(not in LMFDB)