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

• Label: 2.29.b_abc
• 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:	$1 + x - 28 x^{2} + 29 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.196263626344$, $\pm0.862930293010$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-115})\)
Galois group:	$C_2^2$
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$844$	$661696$	$599074576$	$501356472064$	$420790525060924$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$785$	$24562$	$708849$	$20515211$	$594906086$	$17249717159$	$500247453889$	$14507134663258$	$420707208769625$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{3}}$.

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-115})\).

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

The base change of $A$ to $\F_{29^{3}}$ is 1.24389.di^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$

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.ab_abc	$2$	(not in LMFDB)
2.29.ac_ch	$3$	(not in LMFDB)
2.29.a_cf	$6$	(not in LMFDB)