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

• Label: 2.31.o_eh
• 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:	$( 1 + 7 x + 31 x^{2} )^{2}$
	$1 + 14 x + 111 x^{2} + 434 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.716379308692$, $\pm0.716379308692$
Angle rank:	$1$ (numerical)
Jacobians:	$8$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1521$	$950625$	$869306256$	$856133825625$	$819472437292041$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$988$	$29176$	$927028$	$28623706$	$887433118$	$27513276886$	$852888585508$	$26439618778216$	$819628386667948$

Jacobians and polarizations

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

• $y^2=7 x^6+30 x^5+24 x^4+10 x^3+24 x^2+30 x+7$
• $y^2=x^6+4 x^5+8 x^4+x^3+8 x^2+4 x+1$
• $y^2=10 x^6+21 x^5+x^4+4 x^3+x^2+21 x+10$
• $y^2=x^6+19 x^3+8$
• $y^2=12 x^6+29 x^5+19 x^4+28 x^3+19 x^2+29 x+12$
• $y^2=14 x^6+30 x^4+18 x^3+30 x^2+14$
• $y^2=9 x^6+6 x^5+28 x^4+8 x^3+28 x^2+6 x+9$
• $y^2=27 x^6+17 x^5+25 x^4+4 x^3+20 x^2+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.h^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.ao_eh	$2$	(not in LMFDB)
2.31.a_n	$2$	(not in LMFDB)
2.31.aw_hb	$3$	(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)