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

• Label: 2.29.au_gc
• 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:	$( 1 - 10 x + 29 x^{2} )^{2}$
	$1 - 20 x + 158 x^{2} - 580 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.121118941591$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$400$	$640000$	$588547600$	$500131840000$	$420828298810000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$758$	$24130$	$707118$	$20517050$	$594887078$	$17250342770$	$500249228638$	$14507160605290$	$420707297939798$

Jacobians and polarizations

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

• $y^2=8 x^6+26 x^4+26 x^2+8$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_abq	$2$	(not in LMFDB)
2.29.u_gc	$2$	(not in LMFDB)
2.29.k_ct	$3$	(not in LMFDB)