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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 19 x + 147 x^{2} - 551 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0535929210309$, $\pm0.216109662409$
Angle rank:	$2$ (numerical)
Number field:	4.0.3725.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$419$	$653221$	$591589871$	$500399293829$	$420763339919824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$775$	$24257$	$707499$	$20513886$	$594826471$	$17249765369$	$500245167219$	$14507138049203$	$420707202040950$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=8x^6+19x^5+22x^4+15x^3+24x^2+4x+5$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is 4.0.3725.1.

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.t_fr	$2$	(not in LMFDB)