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

• Label: 2.29.ar_ez
• 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:	$1 - 17 x + 129 x^{2} - 493 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.148587705549$, $\pm0.259628098963$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-170 +34 \sqrt{5}})\)
Galois group:	$C_4$
Jacobians:	$2$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$461$	$682741$	$599420321$	$501777084245$	$420945811577296$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$811$	$24577$	$709443$	$20522778$	$594860491$	$17249916277$	$500246212803$	$14507145962593$	$420707247339606$

Jacobians and polarizations

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

• $y^2=3 x^6+5 x^5+20 x^4+16 x^3+9 x^2+x+14$
• $y^2=17 x^6+26 x^5+8 x^4+2 x^3+8 x^2+14 x+15$

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 \(\Q(\sqrt{-170 +34 \sqrt{5}})\).

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.r_ez	$2$	(not in LMFDB)