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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 + x + 5 x^{2} )^{4}$
	$1 + 4 x + 26 x^{2} + 64 x^{3} + 211 x^{4} + 320 x^{5} + 650 x^{6} + 500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.571783146564$, $\pm0.571783146564$, $\pm0.571783146564$, $\pm0.571783146564$
Angle rank:	$1$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2401$	$1500625$	$157351936$	$125333700625$	$108441586233841$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$62$	$70$	$502$	$3530$	$15842$	$75890$	$391782$	$1963150$	$9749822$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.b^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-19}) \)$)$

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.ae_ba_acm_id	$2$	(not in LMFDB)
4.5.ac_u_abc_ft	$2$	(not in LMFDB)
4.5.a_s_a_fb	$2$	(not in LMFDB)
4.5.c_u_bc_ft	$2$	(not in LMFDB)
4.5.ac_ah_ac_cy	$3$	(not in LMFDB)
4.5.b_f_ao_ao	$3$	(not in LMFDB)