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

• Label: 2.29.af_w
• 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:	$( 1 - 9 x + 29 x^{2} )( 1 + 4 x + 29 x^{2} )$
	$1 - 5 x + 22 x^{2} - 145 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.185103371333$, $\pm0.621118941591$
Angle rank:	$2$ (numerical)
Jacobians:	$30$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$714$	$723996$	$589247064$	$501005232000$	$421063731011154$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$861$	$24160$	$708353$	$20528525$	$594837306$	$17249912705$	$500247905953$	$14507140055440$	$420707164368981$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.aj $\times$ 1.29.e 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.aj : \(\Q(\sqrt{-35}) \).
• 1.29.e : \(\Q(\sqrt{-1}) \).

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.an_dq	$2$	(not in LMFDB)
2.29.f_w	$2$	(not in LMFDB)
2.29.n_dq	$2$	(not in LMFDB)