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

• Label: 2.29.e_ck
• 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:	$( 1 + 2 x + 29 x^{2} )^{2}$
	$1 + 4 x + 62 x^{2} + 116 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.559453748998$, $\pm0.559453748998$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1024$	$802816$	$586802176$	$498503778304$	$421006051738624$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$950$	$24058$	$704814$	$20525714$	$594865766$	$17249369066$	$500246196574$	$14507161118722$	$420707209289750$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.ae_ck	$2$	(not in LMFDB)
2.29.a_cc	$2$	(not in LMFDB)
2.29.ac_az	$3$	(not in LMFDB)