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

• Label: 2.29.ap_ei
• 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:	$( 1 - 9 x + 29 x^{2} )( 1 - 6 x + 29 x^{2} )$
	$1 - 15 x + 112 x^{2} - 435 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.185103371333$, $\pm0.311919362152$
Angle rank:	$2$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$504$	$707616$	$603669024$	$501912028800$	$420853401410904$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$841$	$24750$	$709633$	$20518275$	$594824326$	$17249813295$	$500246477473$	$14507148445830$	$420707234868481$

Jacobians and polarizations

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

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

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.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.aj : \(\Q(\sqrt{-35}) \).
• 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.ad_e	$2$	(not in LMFDB)
2.29.d_e	$2$	(not in LMFDB)
2.29.p_ei	$2$	(not in LMFDB)