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

• Label: 2.31.ah_ct
• Base field: $\F_{31}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$1 - 7 x + 71 x^{2} - 217 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.342012631269$, $\pm0.451294890566$
Angle rank:	$2$ (numerical)
Number field:	4.0.1962597.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$809$	$1016913$	$902422511$	$852153772653$	$819228386350544$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$1055$	$30289$	$922723$	$28615180$	$887481083$	$27512796931$	$852891657619$	$26439622044871$	$819628303526990$

Jacobians and polarizations

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

• $y^2=17 x^6+25 x^5+21 x^4+29 x^3+23 x^2+10 x+26$
• $y^2=14 x^6+26 x^5+26 x^4+13 x^3+21 x^2+18 x+6$
• $y^2=5 x^6+21 x^5+23 x^4+20 x^3+30 x^2+21 x+7$
• $y^2=15 x^6+8 x^5+2 x^4+3 x^3+18 x^2+28 x+24$
• $y^2=26 x^6+22 x^5+4 x^4+4 x^3+30 x^2+17 x+26$
• $y^2=11 x^6+18 x^5+5 x^4+30 x^3+24 x^2+3 x+16$
• $y^2=30 x^6+19 x^5+29 x^4+13 x^3+6 x^2+19 x+6$
• $y^2=16 x^6+23 x^5+30 x^4+12 x^3+5 x^2+15 x+1$
• $y^2=9 x^6+17 x^5+16 x^3+4 x^2+3 x+13$
• $y^2=19 x^6+29 x^5+13 x^4+26 x^3+17 x^2+26 x+23$
• $y^2=3 x^6+26 x^5+16 x^4+19 x^3+19 x^2+5 x+22$
• $y^2=10 x^6+4 x^5+23 x^4+17 x^3+14 x+19$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is 4.0.1962597.1.

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.h_ct	$2$	(not in LMFDB)