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

• Label: 2.29.a_cf
• 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:	$( 1 - x + 29 x^{2} )( 1 + x + 29 x^{2} )$
	$1 + 57 x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.470403040323$, $\pm0.529596959677$
Angle rank:	$1$ (numerical)
Jacobians:	$10$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$899$	$808201$	$594864704$	$498033661225$	$420707257830779$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$956$	$24390$	$704148$	$20511150$	$594906086$	$17249876310$	$500244331108$	$14507145975870$	$420707282361356$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{2}}$.

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ab $\times$ 1.29.b 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.ab : \(\Q(\sqrt{-115}) \).
• 1.29.b : \(\Q(\sqrt{-115}) \).

Endomorphism algebra over $\overline{\F}_{29}$

The base change of $A$ to $\F_{29^{2}}$ is 1.841.cf^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$

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.ac_ch	$2$	(not in LMFDB)
2.29.c_ch	$2$	(not in LMFDB)
2.29.a_acf	$4$	(not in LMFDB)