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

• Label: 2.29.a_cg
• Base field: $\F_{29}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• 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:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$36$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 5$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$900$	$810000$	$594872100$	$497871360000$	$420707274322500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$958$	$24390$	$703918$	$20511150$	$594920878$	$17249876310$	$500243583838$	$14507145975870$	$420707315344798$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{2}}$.

Endomorphism algebra over $\F_{29}$

The isogeny class factors as 1.29.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-29}) \)$)$

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

The base change of $A$ to $\F_{29^{2}}$ is 1.841.cg^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_abd	$3$	(not in LMFDB)
2.29.a_acg	$4$	(not in LMFDB)
2.29.a_a	$8$	(not in LMFDB)