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

• Label: 2.29.a_abh
• 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:	$1 - 33 x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.153671890037$, $\pm0.846328109963$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{91})\)
Galois group:	$C_2^2$
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$809$	$654481$	$594870644$	$501087015625$	$420707228578529$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$776$	$24390$	$708468$	$20511150$	$594917966$	$17249876310$	$500248538788$	$14507145975870$	$420707223856856$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{2}}$.

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{91})\).

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.abh^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$

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.ak_df	$4$	(not in LMFDB)
2.29.a_bh	$4$	(not in LMFDB)
2.29.k_df	$4$	(not in LMFDB)