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

• Label: 2.29.at_fs
• 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: no

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 29 x^{2} )( 1 - 9 x + 29 x^{2} )$
	$1 - 19 x + 148 x^{2} - 551 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.185103371333$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	4

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$420$	$655200$	$593011440$	$501005232000$	$420948531950100$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$777$	$24314$	$708353$	$20522911$	$594901062$	$17250266779$	$500247905953$	$14507149497026$	$420707229008577$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ak $\times$ 1.29.aj 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.ak : \(\Q(\sqrt{-1}) \).
• 1.29.aj : \(\Q(\sqrt{-35}) \).

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.ab_abg	$2$	(not in LMFDB)
2.29.b_abg	$2$	(not in LMFDB)
2.29.t_fs	$2$	(not in LMFDB)