Abelian variety isogeny class 2.29.aq_eo over $\F_{29}$

• Label: 2.29.aq_eo
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 29 x^{2} )( 1 - 6 x + 29 x^{2} )$
	$1 - 16 x + 118 x^{2} - 464 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.311919362152$
Angle rank:	$2$ (numerical)
Jacobians:	$22$
Isomorphism classes:	104
Cyclic group of points:	no
Non-cyclic primes:	$2$

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$480$	$691200$	$599124960$	$501037056000$	$420733195442400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$822$	$24566$	$708398$	$20512414$	$594810342$	$17249889286$	$500247800158$	$14507159554094$	$420707303799702$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):

• $y^2=14 x^6+11 x^5+6 x^4+10 x^3+21 x^2+22 x+4$
• $y^2=14 x^6+7 x^5+26 x^4+11 x^3+3 x^2+7 x+15$
• $y^2=8 x^6+7 x^5+17 x^4+19 x^3+17 x^2+7 x+8$
• $y^2=28 x^6+19 x^5+21 x^4+13 x^3+21 x^2+19 x+28$
• $y^2=26 x^6+14 x^5+16 x^4+17 x^3+13 x^2+22 x+27$
• $y^2=25 x^6+10 x^5+25 x^4+24 x^3+25 x^2+10 x+25$
• $y^2=27 x^6+22 x^5+12 x^4+22 x^3+12 x^2+22 x+27$
• $y^2=22 x^6+6 x^5+22 x^3+9 x^2+28 x+14$
• $y^2=23 x^6+2 x^5+7 x^4+5 x^3+7 x^2+2 x+23$
• $y^2=13 x^6+x^5+25 x^4+14 x^3+15 x^2+18 x+17$
• $y^2=26 x^6+x^5+23 x^4+23 x^3+11 x^2+2 x+10$
• $y^2=19 x^6+12 x^5+19 x^4+15 x^3+19 x^2+12 x+19$
• $y^2=17 x^6+22 x^5+21 x^4+27 x^3+3 x^2+4 x+14$
• $y^2=11 x^6+6 x^5+26 x^4+24 x^3+11 x^2+20 x+28$
• $y^2=19 x^6+14 x^5+3 x^4+14 x^3+7 x^2+18 x+15$
• $y^2=8 x^6+x^5+10 x^4+7 x^3+10 x^2+x+8$
• $y^2=27 x^6+13 x^5+8 x^4+22 x^3+8 x^2+13 x+27$
• $y^2=17 x^6+27 x^5+10 x^4+23 x^3+10 x^2+27 x+17$
• $y^2=2 x^6+4 x^5+10 x^4+3 x^3+15 x^2+9 x+14$
• $y^2=19 x^6+18 x^5+14 x^4+16 x^3+21 x^2+26 x+17$
• $y^2=4 x^6+3 x^5+9 x^4+6 x^3+9 x^2+3 x+4$
• $y^2=8 x^6+7 x^5+27 x^4+23 x^3+16 x^2+28 x+17$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29}$.

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ak $\times$ 1.29.ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.29.ak : \(\Q(\sqrt{-1}) \).
• 1.29.ag : \(\Q(\sqrt{-5}) \).

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
2.29.ae_ac	$2$	(not in LMFDB)
2.29.e_ac	$2$	(not in LMFDB)
2.29.q_eo	$2$	(not in LMFDB)