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

• Label: 2.31.k_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:	$\pm0.648224405710$, $\pm0.648224405710$
Angle rank:	$1$ (numerical)
Jacobians:	$16$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1369$	$998001$	$867420304$	$853914605625$	$820073588003329$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$42$	$1036$	$29112$	$924628$	$28644702$	$887391646$	$27512692242$	$852894119908$	$26439604326312$	$819628280596156$

Jacobians and polarizations

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

• $y^2=3 x^6+21 x^3+6$
• $y^2=17 x^6+27 x^5+4 x^4+7 x^3+4 x^2+27 x+17$
• $y^2=29 x^6+2 x^5+18 x^4+13 x^3+29 x^2+11 x+17$
• $y^2=9 x^6+11 x^5+27 x^4+24 x^3+27 x^2+11 x+9$
• $y^2=16 x^6+12 x^5+27 x^4+19 x^3+27 x^2+12 x+16$
• $y^2=3 x^6+13 x^3+3$
• $y^2=3 x^6+3 x^3+3$
• $y^2=7 x^6+20 x^5+25 x^4+7 x^3+25 x^2+20 x+7$
• $y^2=3 x^6+2 x^5+10 x^4+29 x^3+27 x^2+9$
• $y^2=21 x^6+8 x^5+25 x^4+23 x^3+25 x^2+8 x+21$
• $y^2=16 x^6+15 x^5+15 x^4+11 x^3+22 x^2+24 x+18$
• $y^2=18 x^6+25 x^5+11 x^4+5 x^3+11 x^2+25 x+18$
• $y^2=26 x^6+7 x^5+4 x^4+23 x^3+4 x^2+7 x+26$
• $y^2=22 x^6+13 x^5+5 x^4+22 x^3+5 x^2+13 x+22$
• $y^2=9 x^6+22 x^5+23 x^4+26 x^3+24 x^2+16 x+5$
• $y^2=30 x^6+5 x^5+2 x^4+27 x^3+2 x^2+5 x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.f^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.ak_dj	$2$	(not in LMFDB)
2.31.a_bl	$2$	(not in LMFDB)
2.31.af_ag	$3$	(not in LMFDB)