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

• Label: 2.31.ai_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:	$\pm0.383045975359$, $\pm0.383045975359$
Angle rank:	$1$ (numerical)
Jacobians:	$25$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$784$	$1016064$	$906010000$	$852534595584$	$819037316093584$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1054$	$30408$	$923134$	$28608504$	$887433118$	$27512971944$	$852894656254$	$26439625543128$	$819628188327454$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.ae^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_bu	$2$	(not in LMFDB)
2.31.i_da	$2$	(not in LMFDB)
2.31.ao_eh	$3$	(not in LMFDB)
2.31.al_dm	$3$	(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)