Abelian variety isogeny class 2.31.ak_dj over $\F_{31}$

• Label: 2.31.ak_dj
• Base field: $\F_{31}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 31 x^{2} )^{2}$
	$1 - 10 x + 87 x^{2} - 310 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.351775594290$, $\pm0.351775594290$
Angle rank:	$1$ (numerical)
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$729$	$998001$	$907937424$	$853914605625$	$819183221376129$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1036$	$30472$	$924628$	$28613602$	$887391646$	$27512535982$	$852894119908$	$26439639995032$	$819628280596156$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=x^6+7 x^3+2$
• $y^2=20 x^6+19 x^5+12 x^4+21 x^3+12 x^2+19 x+20$
• $y^2=25 x^6+6 x^5+23 x^4+8 x^3+25 x^2+2 x+20$
• $y^2=27 x^6+2 x^5+19 x^4+10 x^3+19 x^2+2 x+27$
• $y^2=17 x^6+5 x^5+19 x^4+26 x^3+19 x^2+5 x+17$
• $y^2=x^6+25 x^3+1$
• $y^2=x^6+x^3+1$
• $y^2=21 x^6+29 x^5+13 x^4+21 x^3+13 x^2+29 x+21$
• $y^2=x^6+11 x^5+24 x^4+20 x^3+9 x^2+3$
• $y^2=x^6+24 x^5+13 x^4+7 x^3+13 x^2+24 x+1$
• $y^2=17 x^6+14 x^5+14 x^4+2 x^3+4 x^2+10 x+23$
• $y^2=23 x^6+13 x^5+2 x^4+15 x^3+2 x^2+13 x+23$
• $y^2=16 x^6+21 x^5+12 x^4+7 x^3+12 x^2+21 x+16$
• $y^2=4 x^6+8 x^5+15 x^4+4 x^3+15 x^2+8 x+4$
• $y^2=3 x^6+28 x^5+18 x^4+19 x^3+8 x^2+26 x+12$
• $y^2=28 x^6+15 x^5+6 x^4+19 x^3+6 x^2+15 x+28$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31}$.

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
2.31.a_bl	$2$	(not in LMFDB)
2.31.k_dj	$2$	(not in LMFDB)
2.31.f_ag	$3$	(not in LMFDB)