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

• Label: 2.31.aq_es
• 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:	$( 1 - 10 x + 31 x^{2} )( 1 - 6 x + 31 x^{2} )$
	$1 - 16 x + 122 x^{2} - 496 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.145000771013$, $\pm0.318871840175$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$572$	$912912$	$895642748$	$854485632000$	$819737800390652$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$950$	$30064$	$925246$	$28632976$	$887500982$	$27512698096$	$852892950526$	$26439637509904$	$819628343423030$

Jacobians and polarizations

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

• $y^2=26 x^6+15 x^5+5 x^4+7 x^3+22 x^2+19 x+5$
• $y^2=2 x^6+24 x^5+2 x^4+17 x^3+14 x^2+21 x+12$
• $y^2=14 x^6+10 x^5+9 x^4+26 x^3+11 x^2+12 x+7$
• $y^2=29 x^6+14 x^5+8 x^4+23 x^3+19 x^2+20 x+9$
• $y^2=28 x^6+5 x^5+25 x^4+6 x^3+28 x^2+15 x+22$
• $y^2=7 x^6+17 x^5+22 x^4+29 x^3+22 x^2+17 x+7$
• $y^2=23 x^6+27 x^5+7 x^4+27 x^3+5 x^2+22 x+27$
• $y^2=29 x^6+16 x^5+29 x^4+16 x^3+29 x^2+16 x+29$
• $y^2=12 x^6+28 x^5+3 x^4+22 x^3+17 x^2+7 x+17$
• $y^2=9 x^6+21 x^5+27 x^4+2 x^3+28 x^2+24 x$
• $y^2=29 x^5+26 x^4+x^3+26 x^2+29 x$
• $y^2=2 x^6+25 x^5+6 x^4+23 x^3+6 x^2+25 x+2$
• $y^2=23 x^6+3 x^5+4 x^4+5 x^3+11 x^2+24 x+20$
• $y^2=23 x^6+14 x^5+23 x^4+14 x^3+23 x^2+14 x+23$
• $y^2=12 x^6+14 x^5+25 x^4+22 x^3+22 x^2+26 x+16$
• $y^2=18 x^6+4 x^5+8 x^4+18 x^3+x^2+2 x+5$

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.ag 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.ag : \(\Q(\sqrt{-22}) \).

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.ae_c	$2$	(not in LMFDB)
2.31.e_c	$2$	(not in LMFDB)
2.31.q_es	$2$	(not in LMFDB)