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

• Label: 2.29.ag_h
• 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:	$1 - 6 x + 7 x^{2} - 174 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.0214139711812$, $\pm0.645252695485$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-5})\)
Galois group:	$C_2^2$
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$669$	$688401$	$580039056$	$499399817049$	$420672633832749$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$820$	$23778$	$706084$	$20509464$	$594733606$	$17249656056$	$500246433604$	$14507133448842$	$420707195120500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{-5})\).

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

The base change of $A$ to $\F_{29^{3}}$ is 1.24389.alu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

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.g_h	$2$	(not in LMFDB)
2.29.m_dq	$3$	(not in LMFDB)
2.29.am_dq	$6$	(not in LMFDB)