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

• Label: 2.29.a_aby
• 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:	$1 - 50 x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.0845842056149$, $\pm0.915415794385$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{3})\)
Galois group:	$C_2^2$
Jacobians:	$16$
Isomorphism classes:	72

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$792$	$627264$	$594824472$	$499091383296$	$420707269604952$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$742$	$24390$	$705646$	$20511150$	$594825622$	$17249876310$	$500247903838$	$14507145975870$	$420707305909702$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-2}, \sqrt{3})\).

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.aby^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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.a_by	$4$	(not in LMFDB)
2.29.ae_i	$8$	(not in LMFDB)
2.29.e_i	$8$	(not in LMFDB)