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

• Label: 2.31.au_gf
• 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:	$( 1 - 11 x + 31 x^{2} )( 1 - 9 x + 31 x^{2} )$
	$1 - 20 x + 161 x^{2} - 620 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0497126420257$, $\pm0.200429117370$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$483$	$851529$	$881571600$	$852892297929$	$819717013272003$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$884$	$29592$	$923524$	$28632252$	$887516318$	$27512561172$	$852889864324$	$26439612076872$	$819628229210804$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=17 x^6+15 x^5+25 x^4+23 x^3+28 x^2+27 x+6$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.al $\times$ 1.31.aj 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.al : \(\Q(\sqrt{-3}) \).
• 1.31.aj : \(\Q(\sqrt{-43}) \).

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.ac_abl	$2$	(not in LMFDB)
2.31.c_abl	$2$	(not in LMFDB)
2.31.u_gf	$2$	(not in LMFDB)
2.31.af_ba	$3$	(not in LMFDB)
2.31.ac_ab	$3$	(not in LMFDB)