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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$1 - 2 x + 54 x^{2} - 58 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.402859355013$, $\pm0.536611834654$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-110 +2 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$36$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$836$	$799216$	$598309316$	$498774721280$	$420622682781796$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$946$	$24532$	$705198$	$20507028$	$594848386$	$17249908732$	$500246637918$	$14507149595308$	$420707209193106$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-110 +2 \sqrt{5}})\).

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.c_cc	$2$	(not in LMFDB)