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

• Label: 4.5.am_cs_akb_bak
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$
	$1 - 12 x + 70 x^{2} - 261 x^{3} + 686 x^{4} - 1305 x^{5} + 1750 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0878807261908$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.450170915301$
Angle rank:	$4$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$54$	$335340$	$265153824$	$152646768000$	$96425619575904$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$22$	$135$	$626$	$3159$	$15955$	$78870$	$390882$	$1953099$	$9774937$

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 $\F_{5}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 1.5.ad $\times$ 2.5.af_n 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 : \(\Q(\sqrt{-11}) \).
• 2.5.af_n : 4.0.4901.1.

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.ag_q_abh_cw	$2$	(not in LMFDB)
4.5.ae_g_d_aba	$2$	(not in LMFDB)
4.5.ac_a_j_ao	$2$	(not in LMFDB)
4.5.c_a_aj_ao	$2$	(not in LMFDB)
4.5.e_g_ad_aba	$2$	(not in LMFDB)
4.5.g_q_bh_cw	$2$	(not in LMFDB)
4.5.m_cs_kb_bak	$2$	(not in LMFDB)