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

• Label: 2.31.aw_hb
• 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: no

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x + 31 x^{2} )^{2}$
	$1 - 22 x + 183 x^{2} - 682 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0497126420257$, $\pm0.0497126420257$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$3, 7$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$441$	$815409$	$869306256$	$850015773369$	$819193181645601$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$844$	$29176$	$920404$	$28613950$	$887433118$	$27512309170$	$852889870564$	$26439618778216$	$819628285947004$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.al^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.31.a_ach	$2$	(not in LMFDB)
2.31.w_hb	$2$	(not in LMFDB)
2.31.ah_s	$3$	(not in LMFDB)
2.31.ae_ap	$3$	(not in LMFDB)
2.31.i_da	$3$	(not in LMFDB)
2.31.l_dm	$3$	(not in LMFDB)
2.31.o_eh	$3$	(not in LMFDB)