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

• Label: 2.29.a_abu
• 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:	$1 - 46 x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.104231981117$, $\pm0.895768018883$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{26})\)
Galois group:	$C_2^2$
Jacobians:	$10$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$796$	$633616$	$594842044$	$499634095104$	$420707273960476$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$750$	$24390$	$706414$	$20511150$	$594860766$	$17249876310$	$500248865374$	$14507145975870$	$420707314620750$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{26})\).

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.abu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-78}) \)$)$

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.ag_bp	$3$	(not in LMFDB)
2.29.g_bp	$3$	(not in LMFDB)
2.29.a_bu	$4$	(not in LMFDB)