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

• Label: 2.29.ao_dv
• 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:	$1 - 14 x + 99 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.134112930066$, $\pm0.373400581930$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-59 +28 \sqrt{2}})\)
Galois group:	$D_{4}$
Jacobians:	$15$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$521$	$709081$	$599625152$	$500407679753$	$420632288294641$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$844$	$24586$	$707508$	$20507496$	$594827686$	$17250226072$	$500249204004$	$14507155268818$	$420707225421564$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

• $y^2=22 x^6+16 x^5+22 x^4+22 x^3+6 x^2+17 x+7$
• $y^2=7 x^6+9 x^5+17 x^4+24 x^2+17 x+10$
• $y^2=7 x^6+17 x^5+22 x^4+12 x^3+25 x^2+7 x+19$
• $y^2=8 x^6+9 x^5+18 x^4+13 x^3+2 x^2+6 x+14$
• $y^2=8 x^6+16 x^5+26 x^4+25 x^3+26 x^2+10 x+2$
• $y^2=16 x^6+18 x^5+14 x^4+6 x^3+14 x+27$
• $y^2=11 x^6+7 x^5+x^4+21 x^3+7 x^2+18 x+1$
• $y^2=4 x^6+17 x^5+x^4+20 x^3+23 x^2+13 x+11$
• $y^2=28 x^6+26 x^5+20 x^3+15 x^2+10 x+27$
• $y^2=2 x^6+26 x^5+27 x^4+4 x^3+26 x^2+27 x+24$
• $y^2=10 x^6+18 x^5+14 x^4+14 x^3+4 x^2+12 x+8$
• $y^2=16 x^6+3 x^5+14 x^4+14 x^3+18 x^2+13 x+19$
• $y^2=15 x^6+25 x^5+5 x^4+8 x^3+22 x^2+2 x+19$
• $y^2=26 x^6+12 x^5+25 x^4+8 x^3+22 x^2+11 x+2$
• $y^2=11 x^6+4 x^5+18 x^4+24 x^3+22 x^2+14 x+3$

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 \(\Q(\sqrt{-59 +28 \sqrt{2}})\).

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_dv	$2$	(not in LMFDB)