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

• Label: 2.31.as_fk
• 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:	$1 - 18 x + 140 x^{2} - 558 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0859487803773$, $\pm0.273644155370$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-40 -18 \sqrt{3}})\)
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$526$	$882628$	$889102534$	$853751942352$	$819684504550006$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$918$	$29846$	$924454$	$28631114$	$887480934$	$27512398130$	$852890533630$	$26439628750670$	$819628377477318$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=3 x^6+3 x^5+22 x^4+4 x^3+6 x^2+14 x+2$
• $y^2=26 x^6+26 x^4+x^3+2 x^2+23 x+27$

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 \(\Q(\sqrt{-40 -18 \sqrt{3}})\).

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.s_fk	$2$	(not in LMFDB)