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

• Label: 2.29.ae_ck
• 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:	$( 1 - 2 x + 29 x^{2} )^{2}$
	$1 - 4 x + 62 x^{2} - 116 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.440546251002$, $\pm0.440546251002$
Angle rank:	$1$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$784$	$802816$	$602997136$	$498503778304$	$420408602961424$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$950$	$24722$	$704814$	$20496586$	$594865766$	$17250383554$	$500246196574$	$14507130833018$	$420707209289750$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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_cc	$2$	(not in LMFDB)
2.29.e_ck	$2$	(not in LMFDB)
2.29.c_az	$3$	(not in LMFDB)