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

• Label: 2.29.ai_cw
• 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:	$( 1 - 4 x + 29 x^{2} )^{2}$
	$1 - 8 x + 74 x^{2} - 232 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.378881058409$, $\pm0.378881058409$
Angle rank:	$1$ (numerical)
Jacobians:	$9$
Cyclic group of points:	no
Non-cyclic primes:	$2, 13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$676$	$781456$	$608806276$	$500131840000$	$420356032687396$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$926$	$24958$	$707118$	$20494022$	$594759566$	$17250117998$	$500249228638$	$14507150229622$	$420707168660606$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_bq	$2$	(not in LMFDB)
2.29.i_cw	$2$	(not in LMFDB)
2.29.e_an	$3$	(not in LMFDB)