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

• Label: 2.31.a_ach
• 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:	$( 1 - 11 x + 31 x^{2} )( 1 + 11 x + 31 x^{2} )$
	$1 - 59 x^{2} + 961 x^{4}$
Frobenius angles:	$\pm0.0497126420257$, $\pm0.950287357974$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$903$	$815409$	$887468400$	$850015773369$	$819628286463903$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$844$	$29792$	$920404$	$28629152$	$887433118$	$27512614112$	$852889870564$	$26439622160672$	$819628285947004$

Jacobians and polarizations

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

• $y^2=x^6+6 x^3+30$
• $y^2=x^6+x^3+30$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{2}}$.

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.al $\times$ 1.31.l 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.al : \(\Q(\sqrt{-3}) \).
• 1.31.l : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{31}$

The base change of $A$ to $\F_{31^{2}}$ is 1.961.ach^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.aw_hb	$2$	(not in LMFDB)
2.31.w_hb	$2$	(not in LMFDB)
2.31.as_fj	$3$	(not in LMFDB)
2.31.ap_ec	$3$	(not in LMFDB)
2.31.ad_bi	$3$	(not in LMFDB)
2.31.a_n	$3$	(not in LMFDB)
2.31.a_bu	$3$	(not in LMFDB)
2.31.d_bi	$3$	(not in LMFDB)
2.31.p_ec	$3$	(not in LMFDB)
2.31.s_fj	$3$	(not in LMFDB)