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

• Label: 2.31.i_da
• 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:	$( 1 + 4 x + 31 x^{2} )^{2}$
	$1 + 8 x + 78 x^{2} + 248 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.616954024641$, $\pm0.616954024641$
Angle rank:	$1$ (numerical)
Jacobians:	$25$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1296$	$1016064$	$869306256$	$852534595584$	$820219585554576$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$1054$	$29176$	$923134$	$28649800$	$887433118$	$27512256280$	$852894656254$	$26439618778216$	$819628188327454$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 25 curves (of which all are hyperelliptic):

• $y^2=8 x^6+30 x^4+30 x^2+8$
• $y^2=4 x^6+21 x^5+15 x^4+29 x^3+15 x^2+21 x+4$
• $y^2=3 x^6+24 x^5+23 x^4+3 x^3+23 x^2+24 x+12$
• $y^2=19 x^6+15 x^5+27 x^4+4 x^3+27 x^2+15 x+19$
• $y^2=3 x^6+24 x^4+24 x^2+3$
• $y^2=21 x^6+27 x^4+27 x^2+21$
• $y^2=x^6+28 x^3+4$
• $y^2=30 x^5+24 x^4+22 x^3+24 x^2+30 x$
• $y^2=6 x^6+16 x^5+18 x^4+22 x^3+3 x^2+28 x$
• $y^2=8 x^6+8 x^5+29 x^4+23 x^3+30 x^2+2 x+1$
• $y^2=10 x^6+30 x^5+26 x^4+5 x^3+24 x^2+3 x+20$
• $y^2=8 x^6+7 x^5+27 x^4+17 x^3+24 x+28$
• $y^2=7 x^6+11 x^5+3 x^4+8 x^3+3 x^2+11 x+7$
• $y^2=11 x^6+21 x^5+20 x^4+26 x^3+20 x^2+21 x+11$
• $y^2=25 x^6+15 x^5+8 x^4+13 x^3+7 x^2+11 x+7$
• $y^2=24 x^6+28 x^5+3 x^4+28 x^3+19 x^2+25 x+17$
• $y^2=10 x^6+29 x^5+6 x^4+12 x^3+6 x^2+29 x+10$
• $y^2=12 x^6+20 x^5+8 x^3+2 x+24$
• $y^2=x^6+9 x^3+16$
• $y^2=12 x^5+25 x^4+18 x^3+24 x^2+16 x+18$
• $y^2=x^6+x^3+2$
• $y^2=x^6+4$
• $y^2=10 x^6+6 x^5+8 x^4+6 x^3+8 x^2+6 x+10$
• $y^2=13 x^6+5 x^5+20 x^4+14 x^3+6 x^2+25 x+15$
• $y^2=x^6+20 x^3+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.e^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.ai_da	$2$	(not in LMFDB)
2.31.a_bu	$2$	(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.l_dm	$3$	(not in LMFDB)
2.31.o_eh	$3$	(not in LMFDB)