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

• Label: 2.31.f_ba
• 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:	$( 1 - 4 x + 31 x^{2} )( 1 + 9 x + 31 x^{2} )$
	$1 + 5 x + 26 x^{2} + 155 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.383045975359$, $\pm0.799570882630$
Angle rank:	$2$ (numerical)
Jacobians:	$32$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1148$	$950544$	$893488400$	$854155036224$	$819026526795428$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$989$	$29992$	$924889$	$28608127$	$887516318$	$27512693497$	$852892257169$	$26439632244472$	$819628180401029$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

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

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

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 $\times$ 1.31.j and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.31.ae : \(\Q(\sqrt{-3}) \).
• 1.31.j : \(\Q(\sqrt{-43}) \).

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.an_du	$2$	(not in LMFDB)
2.31.af_ba	$2$	(not in LMFDB)
2.31.n_du	$2$	(not in LMFDB)
2.31.c_ab	$3$	(not in LMFDB)
2.31.u_gf	$3$	(not in LMFDB)