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

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

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 29 x^{2} )( 1 + 4 x + 29 x^{2} )$
	$1 - 6 x + 18 x^{2} - 174 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.121118941591$, $\pm0.621118941591$
Angle rank:	$1$ (numerical)
Jacobians:	$41$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$680$	$707200$	$584811560$	$500131840000$	$420943464967400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$842$	$23976$	$707118$	$20522664$	$594823322$	$17249988696$	$500249228638$	$14507151163704$	$420707233300202$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 41 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{4}}$.

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.ak $\times$ 1.29.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.29.ak : \(\Q(\sqrt{-1}) \).
• 1.29.e : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{29^{4}}$ is 1.707281.ade^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{29^{2}}$
  The base change of $A$ to $\F_{29^{2}}$ is 1.841.abq $\times$ 1.841.bq. The endomorphism algebra for each factor is:

  • 1.841.abq : \(\Q(\sqrt{-1}) \).
  • 1.841.bq : \(\Q(\sqrt{-1}) \).

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.ao_du	$2$	(not in LMFDB)
2.29.g_s	$2$	(not in LMFDB)
2.29.o_du	$2$	(not in LMFDB)