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

• Label: 2.29.d_cg
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 + 29 x^{2} )( 1 + 3 x + 29 x^{2} )$
	$1 + 3 x + 58 x^{2} + 87 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.589851478136$
Angle rank:	$1$ (numerical)
Jacobians:	$18$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$990$	$801900$	$589164840$	$498550852800$	$420890705536950$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$949$	$24156$	$704881$	$20520093$	$594866122$	$17249634897$	$500245896001$	$14507150284044$	$420707235367549$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

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.a $\times$ 1.29.d 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.a : \(\Q(\sqrt{-29}) \).
• 1.29.d : \(\Q(\sqrt{-107}) \).

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.bx $\times$ 1.841.cg. The endomorphism algebra for each factor is:

• 1.841.bx : \(\Q(\sqrt{-107}) \).
• 1.841.cg : the quaternion algebra over \(\Q\) ramified at $29$ and $\infty$.

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.ad_cg	$2$	(not in LMFDB)