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

• Label: 4.5.am_ct_aki_bbg
• 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:	$( 1 - 3 x + 5 x^{2} )( 1 - x + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{2}$
	$1 - 12 x + 71 x^{2} - 268 x^{3} + 708 x^{4} - 1340 x^{5} + 1775 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.428216853436$
Angle rank:	$3$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$60$	$378000$	$300061440$	$164505600000$	$98719892274300$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$24$	$150$	$672$	$3234$	$16074$	$79290$	$391872$	$1952574$	$9765624$

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^2 $\times$ 1.5.ad $\times$ 1.5.ab 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.5.ad : \(\Q(\sqrt{-11}) \).
• 1.5.ab : \(\Q(\sqrt{-19}) \).

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.ak_bx_agc_pc	$2$	(not in LMFDB)
4.5.ag_r_abi_cu	$2$	(not in LMFDB)
4.5.ae_h_au_cq	$2$	(not in LMFDB)
4.5.ae_h_e_abc	$2$	(not in LMFDB)
4.5.ac_b_c_i	$2$	(not in LMFDB)
4.5.c_b_ac_i	$2$	(not in LMFDB)
4.5.e_h_ae_abc	$2$	(not in LMFDB)
4.5.e_h_u_cq	$2$	(not in LMFDB)
4.5.g_r_bi_cu	$2$	(not in LMFDB)
4.5.k_bx_gc_pc	$2$	(not in LMFDB)
4.5.m_ct_ki_bbg	$2$	(not in LMFDB)
4.5.a_i_i_bh	$3$	(not in LMFDB)