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

• Label: 2.29.ao_dp
• 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:	$1 - 14 x + 93 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0232315260076$, $\pm0.402168915073$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-22 +4 \sqrt{14}})\)
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$515$	$697825$	$593430380$	$498711103625$	$420348118639075$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$832$	$24334$	$705108$	$20493636$	$594765982$	$17249866804$	$500246330148$	$14507137149766$	$420707161799952$

Jacobians and polarizations

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

• $y^2=8 x^6+8 x^5+19 x^4+3 x^3+15 x^2+8 x+14$
• $y^2=11 x^6+19 x^5+27 x^4+23 x^3+20 x^2+5 x+26$

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 \(\Q(\sqrt{-22 +4 \sqrt{14}})\).

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.o_dp	$2$	(not in LMFDB)