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

• Label: 2.29.aj_ce
• 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:	$1 - 9 x + 56 x^{2} - 261 x^{3} + 841 x^{4}$
Frobenius angles:	$\pm0.173080111599$, $\pm0.506413444932$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{89})\)
Galois group:	$C_2^2$
Jacobians:	$24$
Isomorphism classes:	24
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$628$	$733504$	$594871744$	$499736275200$	$420895389603748$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$873$	$24390$	$706561$	$20520321$	$594920166$	$17250046149$	$500245518241$	$14507145975870$	$420707246286753$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{89})\).

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

The base change of $A$ to $\F_{29^{6}}$ is 1.594823321.ctqk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-267}) \)$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{29^{2}}$ is the simple isogeny class 2.841.bf_eq and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{89})\).

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

  The base change of $A$ to $\F_{29^{3}}$ is the simple isogeny class 2.24389.a_ctqk and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{89})\).

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.j_ce	$2$	(not in LMFDB)
2.29.a_abf	$3$	(not in LMFDB)
2.29.j_ce	$3$	(not in LMFDB)