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

• Label: 2.29.aq_en
• 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:	$1 - 16 x + 117 x^{2} - 464 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.100679126603$, $\pm0.320248433638$
Angle rank:	$2$ (numerical)
Number field:	4.0.23225.1
Galois group:	$D_{4}$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$479$	$689281$	$597937616$	$500644779449$	$420651155542399$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$820$	$24518$	$707844$	$20508414$	$594791686$	$17249843366$	$500247845124$	$14507160366782$	$420707307311700$

Jacobians and polarizations

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

• $y^2=27x^6+7x^5+2x^4+19x^3+22x^2+16x+26$
• $y^2=12x^6+13x^5+x^4+9x^3+19x+7$
• $y^2=12x^6+21x^4+6x^3+10x^2+5x+2$
• $y^2=2x^6+26x^4+3x^3+15x^2+26x+13$
• $y^2=16x^6+11x^5+11x^3+27$
• $y^2=12x^6+3x^5+21x^4+6x^3+5x^2+x+18$

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 4.0.23225.1.

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.q_en	$2$	(not in LMFDB)