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

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

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$1 + 5 x - 6 x^{2} + 155 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.314891072377$, $\pm0.981557739044$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$5$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1116$	$888336$	$907937424$	$852379712064$	$819850910459076$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$925$	$30472$	$922969$	$28636927$	$887391646$	$27512653177$	$852889496209$	$26439639995032$	$819628290173125$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+7$
• $y^2=18 x^5+17 x^4+8 x^3+24 x^2+29 x+28$
• $y^2=19 x^6+26 x^4+16 x^3+x^2+30 x+1$
• $y^2=x^6+x^3+25$
• $y^2=x^6+3 x^3+9$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{3}}$.

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{31^{3}}$ is 1.29791.nc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.af_ag	$2$	(not in LMFDB)
2.31.ak_dj	$3$	(not in LMFDB)
2.31.a_bl	$6$	(not in LMFDB)