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

• Label: 2.29.ao_ec
• 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:	$( 1 - 8 x + 29 x^{2} )( 1 - 6 x + 29 x^{2} )$
	$1 - 14 x + 106 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.233506187634$, $\pm0.311919362152$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	48

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$528$	$722304$	$606879504$	$502261309440$	$420833322645648$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$858$	$24880$	$710126$	$20517296$	$594793386$	$17249548304$	$500245097566$	$14507145006160$	$420707251161978$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ai $\times$ 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.ai : \(\Q(\sqrt{-13}) \).
• 1.29.ag : \(\Q(\sqrt{-5}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.29.ac_k	$2$	(not in LMFDB)
2.29.c_k	$2$	(not in LMFDB)
2.29.o_ec	$2$	(not in LMFDB)