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

• Label: 2.29.ao_ed
• 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:	$( 1 - 7 x + 29 x^{2} )^{2}$
	$1 - 14 x + 107 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.274796655058$, $\pm0.274796655058$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$23$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$529$	$724201$	$607918336$	$502515107689$	$420850577417449$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$860$	$24922$	$710484$	$20518136$	$594779366$	$17249366024$	$500244115684$	$14507144693218$	$420707290942700$

Jacobians and polarizations

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

• $y^2=3 x^6+18 x^5+3 x^4+x^3+26 x^2+18 x+26$
• $y^2=4 x^6+19 x^5+9 x^3+21 x^2+22 x+4$
• $y^2=2 x^6+16 x^5+18 x^4+13 x^3+12 x^2+24 x+18$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29}$.

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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.a_j	$2$	(not in LMFDB)
2.29.o_ed	$2$	(not in LMFDB)
2.29.h_u	$3$	(not in LMFDB)