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

• Label: 2.31.ap_en
• 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:	$1 - 15 x + 117 x^{2} - 465 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.218292075685$, $\pm0.305733869223$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-266 -30 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$599$	$933841$	$902364149$	$855888622525$	$819894585474704$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$971$	$30287$	$926763$	$28638452$	$887485511$	$27512284817$	$852889433923$	$26439619248587$	$819628295251406$

Jacobians and polarizations

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

• $y^2=16 x^6+21 x^5+12 x^4+23 x^3+28 x^2+13 x+3$
• $y^2=6 x^6+6 x^4+25 x^3+13 x^2+28 x+14$
• $y^2=10 x^6+4 x^5+21 x^4+14 x^3+5 x^2+19$
• $y^2=17 x^6+25 x^5+9 x^4+29 x^3+15 x+27$
• $y^2=14 x^6+8 x^5+15 x^4+23 x^3+2 x^2+25 x+21$
• $y^2=21 x^6+26 x^5+30 x^4+29 x^3+10 x^2+12 x+2$
• $y^2=15 x^6+14 x^5+2 x^4+23 x^3+x^2+30 x+29$
• $y^2=6 x^6+15 x^5+11 x^4+19 x^3+29 x^2+28 x+13$

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 \(\Q(\sqrt{-266 -30 \sqrt{5}})\).

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.p_en	$2$	(not in LMFDB)