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

• Label: 2.29.ao_du
• 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:	$( 1 - 10 x + 29 x^{2} )( 1 - 4 x + 29 x^{2} )$
	$1 - 14 x + 98 x^{2} - 406 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.378881058409$
Angle rank:	$1$ (numerical)
Jacobians:	$23$
Isomorphism classes:	82

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$520$	$707200$	$598591240$	$500131840000$	$420592099462600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$842$	$24544$	$707118$	$20505536$	$594823322$	$17250230384$	$500249228638$	$14507155417456$	$420707233300202$

Jacobians and polarizations

This isogeny class contains the Jacobians of 23 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{4}}$.

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ak $\times$ 1.29.ae 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.ak : \(\Q(\sqrt{-1}) \).
• 1.29.ae : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{29^{4}}$ is 1.707281.ade^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{29^{2}}$
  The base change of $A$ to $\F_{29^{2}}$ is 1.841.abq $\times$ 1.841.bq. The endomorphism algebra for each factor is:

  • 1.841.abq : \(\Q(\sqrt{-1}) \).
  • 1.841.bq : \(\Q(\sqrt{-1}) \).

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.ag_s	$2$	(not in LMFDB)
2.29.g_s	$2$	(not in LMFDB)
2.29.o_du	$2$	(not in LMFDB)