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

• Label: 2.31.ah_cu
• 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:	$( 1 - 5 x + 31 x^{2} )( 1 - 2 x + 31 x^{2} )$
	$1 - 7 x + 72 x^{2} - 217 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.351775594290$, $\pm0.442517941024$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$810$	$1018980$	$903056040$	$852071076000$	$819165275772750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$1057$	$30310$	$922633$	$28612975$	$887475562$	$27512891305$	$852892346353$	$26439620809210$	$819628270453177$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=5 x^6+17 x^5+21 x^4+14 x^3+12 x^2+23 x+22$
• $y^2=27 x^6+8 x^5+2 x^4+7 x^3+11 x^2+3 x+5$
• $y^2=15 x^6+23 x^5+14 x^4+11 x^3+25 x^2+4 x+16$
• $y^2=5 x^6+15 x^4+5 x^3+27 x^2+x+6$
• $y^2=4 x^6+18 x^5+8 x^4+4 x^3+8 x^2+28 x+26$
• $y^2=22 x^6+6 x^5+19 x^4+7 x^3+2 x^2+13 x+8$
• $y^2=x^6+20 x^5+16 x^4+24 x^3+25 x^2+17 x+9$
• $y^2=19 x^6+23 x^5+18 x^4+7 x^3+26 x^2+4 x+19$
• $y^2=6 x^6+14 x^5+28 x^4+16 x^3+19 x^2+18 x+21$
• $y^2=15 x^6+22 x^4+14 x^3+19 x^2+11 x+15$
• $y^2=27 x^6+x^5+16 x^4+26 x^2+5 x+22$
• $y^2=16 x^6+24 x^5+10 x^4+26 x^3+23 x^2+11 x+2$

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 $\times$ 1.31.ac 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.af : \(\Q(\sqrt{-11}) \).
• 1.31.ac : \(\Q(\sqrt{-30}) \).

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.ad_ca	$2$	(not in LMFDB)
2.31.d_ca	$2$	(not in LMFDB)
2.31.h_cu	$2$	(not in LMFDB)