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

• Label: 2.29.aj_cs
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 70 x^{2} - 261 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.260020945481$, $\pm0.451708829335$
Angle rank:	$2$ (numerical)
Number field:	4.0.7609932.1
Galois group:	$D_{4}$
Jacobians:	$22$
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$642$	$758844$	$604124568$	$500448511872$	$420688596529722$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$901$	$24768$	$707569$	$20510241$	$594844090$	$17249883861$	$500244548545$	$14507134852464$	$420707248203661$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is 4.0.7609932.1.

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.j_cs	$2$	(not in LMFDB)