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

• Label: 2.29.a_acg
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{29}$
Dimension:	$2$
L-polynomial:	$( 1 - 29 x^{2} )^{2}$
	$1 - 58 x^{2} + 841 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{29}) \)
Galois group:	$C_2$
Jacobians:	$4$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$784$	$614656$	$594774544$	$497871360000$	$420707192277904$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$726$	$24390$	$703918$	$20511150$	$594725766$	$17249876310$	$500243583838$	$14507145975870$	$420707151255606$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=24 x^6+18 x^4+9 x^3+20 x^2+26$
• $y^2=x^5+28 x$
• $y^2=12 x^5+20 x^3+3 x$
• $y^2=28 x^6+16 x^5+5 x^4+18 x^3+11 x^2+9 x+10$

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 the quaternion algebra over \(\Q(\sqrt{29}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.acg^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $29$ and $\infty$.

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_bd	$3$	(not in LMFDB)
2.29.a_cg	$4$	(not in LMFDB)
2.29.a_bd	$6$	(not in LMFDB)