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

• Label: 2.29.ao_dq
• 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:	$1 - 14 x + 94 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0557514217637$, $\pm0.397939584956$
Angle rank:	$2$ (numerical)
Number field:	4.0.62192.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$516$	$699696$	$594461412$	$499000796928$	$420402654216516$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$834$	$24376$	$705518$	$20496296$	$594782226$	$17249992832$	$500247360094$	$14507144089888$	$420707199735714$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

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 4.0.62192.1.

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_dq	$2$	(not in LMFDB)