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

• Label: 4.5.ae_m_abk_ec
• 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: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x + 4 x^{2} - 10 x^{3} + 25 x^{4} )^{2}$
	$1 - 4 x + 12 x^{2} - 36 x^{3} + 106 x^{4} - 180 x^{5} + 300 x^{6} - 500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.196618866693$, $\pm0.196618866693$, $\pm0.619957764542$, $\pm0.619957764542$
Angle rank:	$2$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$571536$	$195608196$	$171634546944$	$109520787787524$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$34$	$98$	$698$	$3562$	$15730$	$78010$	$392538$	$1947554$	$9743074$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=2 x^{10}+x^9+3 x^8+x^7+2 x^6+x^4+x^3+4 x^2+3$
• $y^2=2 x^{10}+x^9+2 x^7+4 x^6+3 x^4+x^3+3 x^2+x+1$
• $y^2=2 x^{10}+x^9+2 x^8+4 x^6+4 x^5+3 x^3+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 2.5.ac_e^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.90944.1$)$

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.a_e_a_ba	$2$	(not in LMFDB)
4.5.e_m_bk_ec	$2$	(not in LMFDB)
4.5.c_a_am_abd	$3$	(not in LMFDB)