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

• Label: 2.29.aq_er
• 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:	$( 1 - 9 x + 29 x^{2} )( 1 - 7 x + 29 x^{2} )$
	$1 - 16 x + 121 x^{2} - 464 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.185103371333$, $\pm0.274796655058$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	8
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$483$	$696969$	$602691264$	$502197528105$	$420959674288923$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$828$	$24710$	$710036$	$20523454$	$594847206$	$17249778406$	$500245349476$	$14507141540990$	$420707225510028$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=28 x^6+22 x^5+18 x^4+6 x^3+18 x^2+22 x+28$
• $y^2=22 x^6+19 x^5+5 x^4+2 x^3+5 x^2+19 x+22$
• $y^2=26 x^6+10 x^5+9 x^4+5 x^3+9 x^2+10 x+26$
• $y^2=3 x^6+10 x^5+9 x^4+26 x^3+9 x^2+10 x+3$
• $y^2=15 x^6+12 x^5+6 x^4+13 x^3+6 x^2+12 x+15$
• $y^2=12 x^6+26 x^5+2 x^4+28 x^3+2 x^2+26 x+12$

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.ah 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.ah : \(\Q(\sqrt{-67}) \).

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.ac_af	$2$	(not in LMFDB)
2.29.c_af	$2$	(not in LMFDB)
2.29.q_er	$2$	(not in LMFDB)