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

• Label: 2.29.ao_dz
• 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:	$( 1 - 9 x + 29 x^{2} )( 1 - 5 x + 29 x^{2} )$
	$1 - 14 x + 103 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.185103371333$, $\pm0.346328109963$
Angle rank:	$2$ (numerical)
Jacobians:	$24$
Isomorphism classes:	40

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$525$	$716625$	$603766800$	$501483425625$	$420764365300125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$852$	$24754$	$709028$	$20513936$	$594821862$	$17249971664$	$500247561028$	$14507149291546$	$420707201410452$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.aj $\times$ 1.29.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.29.aj : \(\Q(\sqrt{-35}) \).
• 1.29.af : \(\Q(\sqrt{-91}) \).

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.ae_n	$2$	(not in LMFDB)
2.29.e_n	$2$	(not in LMFDB)
2.29.o_dz	$2$	(not in LMFDB)