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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 8 x + 35 x^{2} - 232 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0998271456989$, $\pm0.566839520968$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-13})\)
Galois group:	$C_2^2$
Jacobians:	$28$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$637$	$711529$	$585930436$	$499084228825$	$420867917456077$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$848$	$24022$	$705636$	$20518982$	$594853166$	$17249768558$	$500247707716$	$14507160442318$	$420707253618128$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{3}}$.

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-13})\).

Endomorphism algebra over $\overline{\F}_{29}$

The base change of $A$ to $\F_{29^{3}}$ is 1.24389.ahc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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.i_bj	$2$	(not in LMFDB)
2.29.q_es	$3$	(not in LMFDB)
2.29.aq_es	$6$	(not in LMFDB)