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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 14 x + 102 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.171987174684$, $\pm0.354155560073$
Angle rank:	$2$ (numerical)
Number field:	4.0.71600.1
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	24

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$524$	$714736$	$602730476$	$501218626304$	$420735646976524$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$850$	$24712$	$708654$	$20512536$	$594826786$	$17250069664$	$500248175454$	$14507151169408$	$420707201824050$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=23x^6+13x^5+22x^4+23x^3+x^2+14x+1$
• $y^2=14x^6+22x^5+11x^4+19x^3+15x^2+24x+23$
• $y^2=10x^6+2x^5+13x^4+6x^3+24x^2+17x+25$
• $y^2=17x^6+19x^5+10x^4+3x^3+x^2+8x+15$
• $y^2=26x^6+12x^5+15x^3+20x^2+13x+21$
• $y^2=4x^6+20x^5+5x^4+6x^3+8x^2+8x+14$
• $y^2=15x^6+26x^5+15x^4+20x^3+10x^2+3x+21$
• $y^2=15x^6+6x^5+27x^4+3x^3+28x^2+7x+3$
• $y^2=12x^6+3x^5+13x^3+24x^2+5x+16$
• $y^2=27x^6+28x^5+6x^4+15x^3+13x^2+4x+3$
• $y^2=25x^6+7x^5+22x^4+13x^3+4x^2+25x+17$
• $y^2=19x^6+2x^5+13x^4+2x^3+19x^2+3x+10$
• $y^2=23x^6+3x^5+12x^4+8x^3+6x^2+18x+14$
• $y^2=15x^6+3x^5+17x^4+27x^3+11x^2+16x+15$
• $y^2=19x^6+x^5+7x^4+x^3+15$
• $y^2=6x^6+28x^5+27x^4+12x^3+24x^2+22x+26$
• $y^2=11x^6+28x^5+23x^4+8x^3+20x^2+5x+10$
• $y^2=4x^6+4x^5+5x^4+20x^3+15x^2+11x+27$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is 4.0.71600.1.

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.o_dy	$2$	(not in LMFDB)