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

• Label: 2.29.af_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:	$( 1 - 5 x + 29 x^{2} )( 1 + 29 x^{2} )$
	$1 - 5 x + 58 x^{2} - 145 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.346328109963$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$750$	$787500$	$602433000$	$499476600000$	$420583694643750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$933$	$24700$	$706193$	$20505125$	$594824778$	$17249814425$	$500246061313$	$14507153085100$	$420707279044173$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

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.af $\times$ 1.29.a 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.af : \(\Q(\sqrt{-91}) \).
• 1.29.a : \(\Q(\sqrt{-29}) \).

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

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

• 1.841.bh : \(\Q(\sqrt{-91}) \).
• 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.f_cg	$2$	(not in LMFDB)