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

• Label: 2.31.ao_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:	$\pm0.283620691308$, $\pm0.283620691308$
Angle rank:	$1$ (numerical)
Jacobians:	$8$
Cyclic group of points:	no
Non-cyclic primes:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$625$	$950625$	$906010000$	$856133825625$	$819784266015625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$988$	$30408$	$927028$	$28634598$	$887433118$	$27511951338$	$852888585508$	$26439625543128$	$819628386667948$

Jacobians and polarizations

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

• $y^2=21 x^6+28 x^5+10 x^4+30 x^3+10 x^2+28 x+21$
• $y^2=21 x^6+22 x^5+13 x^4+21 x^3+13 x^2+22 x+21$
• $y^2=24 x^6+7 x^5+21 x^4+22 x^3+21 x^2+7 x+24$
• $y^2=3 x^6+26 x^3+24$
• $y^2=4 x^6+20 x^5+27 x^4+30 x^3+27 x^2+20 x+4$
• $y^2=15 x^6+10 x^4+6 x^3+10 x^2+15$
• $y^2=27 x^6+18 x^5+22 x^4+24 x^3+22 x^2+18 x+27$
• $y^2=19 x^6+20 x^5+13 x^4+12 x^3+29 x^2+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.ah^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_n	$2$	(not in LMFDB)
2.31.o_eh	$2$	(not in LMFDB)
2.31.al_dm	$3$	(not in LMFDB)
2.31.ai_da	$3$	(not in LMFDB)
2.31.e_ap	$3$	(not in LMFDB)
2.31.h_s	$3$	(not in LMFDB)
2.31.w_hb	$3$	(not in LMFDB)