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

• Label: 2.29.a_ab
• 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:	$1 - x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.247255813287$, $\pm0.752744186713$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-57}, \sqrt{59})\)
Galois group:	$C_2^2$
Jacobians:	$24$
Isomorphism classes:	32
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$841$	$707281$	$594825844$	$502628535369$	$420707229768001$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$840$	$24390$	$710644$	$20511150$	$594828366$	$17249876310$	$500243590564$	$14507145975870$	$420707226235800$

Jacobians and polarizations

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

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

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 \(\Q(\sqrt{-57}, \sqrt{59})\).

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3363}) \)$)$

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_b	$4$	(not in LMFDB)