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

• Label: 2.29.ar_ey
• 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:	$( 1 - 10 x + 29 x^{2} )( 1 - 7 x + 29 x^{2} )$
	$1 - 17 x + 128 x^{2} - 493 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.274796655058$
Angle rank:	$2$ (numerical)
Jacobians:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$460$	$680800$	$598154560$	$501322057600$	$420839437966300$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$809$	$24526$	$708801$	$20517593$	$594833222$	$17249854397$	$500246672161$	$14507152649254$	$420707294441249$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=21 x^6+21 x^5+8 x^4+17 x^3+7 x^2+7 x+21$
• $y^2=18 x^6+12 x^5+x^4+5 x^3+17 x^2+12 x+12$
• $y^2=14 x^6+23 x^5+25 x^4+10 x^3+21 x^2+26 x+14$

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.ah 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.ah : \(\Q(\sqrt{-67}) \).

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.ad_am	$2$	(not in LMFDB)
2.29.d_am	$2$	(not in LMFDB)
2.29.r_ey	$2$	(not in LMFDB)