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

• Label: 2.29.a_bd
• 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:	$1 + 29 x^{2} + 841 x^{4}$
Frobenius angles:	$\pm0.333333333333$, $\pm0.666666666667$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{29})\)
Galois group:	$C_2^2$
Jacobians:	$65$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$871$	$758641$	$594774544$	$501438183129$	$420707253811351$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$900$	$24390$	$708964$	$20511150$	$594725766$	$17249876310$	$500247827524$	$14507145975870$	$420707274322500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 65 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{29^{6}}$.

Endomorphism algebra over $\F_{29}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{29})\).

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

The base change of $A$ to $\F_{29^{6}}$ is 1.594823321.acuec^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $29$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{29^{2}}$

  The base change of $A$ to $\F_{29^{2}}$ is 1.841.bd^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{29^{3}}$

  The base change of $A$ to $\F_{29^{3}}$ is the simple isogeny class 2.24389.a_acuec and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{29}) \) ramified at both real infinite places.

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_acg	$3$	(not in LMFDB)
2.29.a_abd	$4$	(not in LMFDB)
2.29.a_acg	$6$	(not in LMFDB)