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

• Label: 2.31.as_fm
• 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:	$( 1 - 10 x + 31 x^{2} )( 1 - 8 x + 31 x^{2} )$
	$1 - 18 x + 142 x^{2} - 558 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.145000771013$, $\pm0.244865078763$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	32

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$528$	$887040$	$892373328$	$855106560000$	$820060758349008$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$922$	$29954$	$925918$	$28644254$	$887564122$	$27512738834$	$852890824318$	$26439619826414$	$819628286739802$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=30 x^6+17 x^5+30 x^4+21 x^3+30 x^2+17 x+30$
• $y^2=23 x^6+10 x^5+7 x^4+17 x^3+7 x^2+10 x+23$
• $y^2=23 x^6+3 x^5+26 x^4+3 x^3+27 x^2+23 x+15$
• $y^2=30 x^6+27 x^5+25 x^4+28 x^3+25 x^2+27 x+30$
• $y^2=21 x^6+x^5+11 x^4+16 x^3+23 x^2+20 x+13$
• $y^2=30 x^6+30 x^5+4 x^4+3 x^3+2 x^2+23 x+27$
• $y^2=30 x^6+8 x^5+27 x^4+17 x^3+24 x^2+9 x+30$
• $y^2=7 x^6+13 x^5+26 x^4+3 x^3+26 x^2+13 x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.ak $\times$ 1.31.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.31.ak : \(\Q(\sqrt{-6}) \).
• 1.31.ai : \(\Q(\sqrt{-15}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.31.ac_as	$2$	(not in LMFDB)
2.31.c_as	$2$	(not in LMFDB)
2.31.s_fm	$2$	(not in LMFDB)