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

• Label: 2.29.m_dm
• 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:	$( 1 + 4 x + 29 x^{2} )( 1 + 8 x + 29 x^{2} )$
	$1 + 12 x + 90 x^{2} + 348 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.621118941591$, $\pm0.766493812366$
Angle rank:	$2$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1292$	$739024$	$583509836$	$501353881600$	$420722221411052$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$878$	$23922$	$708846$	$20511882$	$594806366$	$17249863218$	$500246526046$	$14507151082218$	$420707180662478$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.e $\times$ 1.29.i 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.e : \(\Q(\sqrt{-1}) \).
• 1.29.i : \(\Q(\sqrt{-13}) \).

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.am_dm	$2$	(not in LMFDB)
2.29.ae_ba	$2$	(not in LMFDB)
2.29.e_ba	$2$	(not in LMFDB)