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

• Label: 2.29.ag_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:	$\pm0.410148521864$, $\pm0.410148521864$
Angle rank:	$1$ (numerical)
Jacobians:	$18$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$729$	$793881$	$606341376$	$499231272969$	$420340491870849$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$940$	$24858$	$705844$	$20493264$	$594811366$	$17250359136$	$500248208164$	$14507137359522$	$420707155390300$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ad^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.a_bx	$2$	(not in LMFDB)
2.29.g_cp	$2$	(not in LMFDB)
2.29.d_au	$3$	(not in LMFDB)