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

• Label: 2.29.as_fg
• 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:	$1 - 18 x + 136 x^{2} - 522 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0268446987164$, $\pm0.264224646566$
Angle rank:	$2$ (numerical)
Number field:	4.0.7488.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$438$	$664884$	$593479926$	$500293295568$	$420636717101238$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$790$	$24336$	$707350$	$20507712$	$594767734$	$17249424204$	$500243983198$	$14507137600956$	$420707223749830$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=27 x^6+8 x^5+16 x^4+11 x^3+2 x^2+x+28$
• $y^2=19 x^5+16 x^4+22 x^3+27 x^2+8 x+27$

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 4.0.7488.1.

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.s_fg	$2$	(not in LMFDB)