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

• Label: 2.29.j_cd
• 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:	$1 + 9 x + 55 x^{2} + 261 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.490487250239$, $\pm0.833005355783$
Angle rank:	$2$ (numerical)
Number field:	4.0.29173077.1
Galois group:	$D_{4}$
Jacobians:	$36$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1167$	$731709$	$595530603$	$499664319957$	$420518254173072$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$871$	$24417$	$706459$	$20501934$	$594920095$	$17249704329$	$500245735795$	$14507144022231$	$420707251628086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

• $y^2=x^6+7 x^4+26 x^3+25 x^2+26 x+26$
• $y^2=x^6+2 x^5+18 x^4+5 x^3+23 x^2+24 x+15$
• $y^2=28 x^6+17 x^5+18 x^4+3 x^3+28 x^2+x+20$
• $y^2=10 x^6+19 x^5+23 x^4+x^3+13 x^2+4 x+23$
• $y^2=4 x^6+10 x^5+23 x^4+27 x^3+22 x^2+4 x+1$
• $y^2=10 x^6+x^5+25 x^4+21 x^3+4 x^2+16 x+24$
• $y^2=17 x^6+2 x^5+28 x^4+6 x^3+28 x^2+19 x+16$
• $y^2=12 x^6+5 x^5+17 x^4+11 x^3+28 x^2+7 x+16$
• $y^2=14 x^6+25 x^5+16 x^4+23 x^3+4 x^2+24 x+5$
• $y^2=5 x^6+4 x^5+11 x^4+25 x^3+11 x^2+9 x+20$
• $y^2=13 x^6+15 x^5+18 x^4+20 x^3+22 x^2+13 x+13$
• $y^2=24 x^6+4 x^5+7 x^4+27 x^3+9 x^2+28 x+28$
• $y^2=7 x^6+25 x^5+19 x^4+5 x^3+17 x^2+25 x$
• $y^2=7 x^6+16 x^5+19 x^4+27 x^3+21 x^2+11 x+16$
• $y^2=18 x^6+3 x^5+11 x^4+13 x^3+7 x^2+2 x+1$
• $y^2=7 x^6+6 x^5+28 x^4+20 x^3+16 x^2+20 x+4$

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 4.0.29173077.1.

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.aj_cd	$2$	(not in LMFDB)