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

• Label: 2.29.a_an
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 13 x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.214021692789$, $\pm0.785978307211$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-5}, \sqrt{71})\)
Galois group:	$C_2^2$
Jacobians:	$26$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$829$	$687241$	$594853924$	$502390352025$	$420707196194029$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$816$	$24390$	$710308$	$20511150$	$594884526$	$17249876310$	$500244663748$	$14507145975870$	$420707159087856$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}, \sqrt{71})\).

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.an^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-355}) \)$)$

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.a_n	$4$	(not in LMFDB)