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

• Label: 2.29.q_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:	$( 1 + 6 x + 29 x^{2} )( 1 + 10 x + 29 x^{2} )$
	$1 + 16 x + 118 x^{2} + 464 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.688080637848$, $\pm0.878881058409$
Angle rank:	$2$ (numerical)
Jacobians:	$22$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1440$	$691200$	$590539680$	$501037056000$	$420681343255200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$822$	$24214$	$708398$	$20509886$	$594810342$	$17249863334$	$500247800158$	$14507132397646$	$420707303799702$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.g $\times$ 1.29.k 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.g : \(\Q(\sqrt{-5}) \).
• 1.29.k : \(\Q(\sqrt{-1}) \).

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