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

• Label: 2.29.g_cp
• 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:	$( 1 + 3 x + 29 x^{2} )^{2}$
	$1 + 6 x + 67 x^{2} + 174 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.589851478136$, $\pm0.589851478136$
Angle rank:	$1$ (numerical)
Jacobians:	$18$
Cyclic group of points:	no
Non-cyclic primes:	$3, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1089$	$793881$	$583512336$	$499231272969$	$421074216728649$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$940$	$23922$	$705844$	$20529036$	$594811366$	$17249393484$	$500248208164$	$14507154592218$	$420707155390300$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.d^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-107}) \)$)$

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.ag_cp	$2$	(not in LMFDB)
2.29.a_bx	$2$	(not in LMFDB)
2.29.ad_au	$3$	(not in LMFDB)