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

• Label: 2.31.f_co
• 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:	$( 1 + x + 31 x^{2} )( 1 + 4 x + 31 x^{2} )$
	$1 + 5 x + 66 x^{2} + 155 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.528623632522$, $\pm0.616954024641$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	56
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1188$	$1031184$	$875674800$	$851052654144$	$820057113807228$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$1069$	$29392$	$921529$	$28644127$	$887519518$	$27512239897$	$852891457489$	$26439627913072$	$819628273280629$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.b $\times$ 1.31.e 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.b : \(\Q(\sqrt{-123}) \).
• 1.31.e : \(\Q(\sqrt{-3}) \).

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.af_co	$2$	(not in LMFDB)
2.31.ad_cg	$2$	(not in LMFDB)
2.31.d_cg	$2$	(not in LMFDB)
2.31.ak_bz	$3$	(not in LMFDB)
2.31.i_cr	$3$	(not in LMFDB)