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

• Label: 2.31.ab_ci
• 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:	$( 1 - 2 x + 31 x^{2} )( 1 + x + 31 x^{2} )$
	$1 - x + 60 x^{2} - 31 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.442517941024$, $\pm0.528623632522$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$990$	$1043460$	$890109000$	$849902343840$	$819520917212250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$1081$	$29878$	$920281$	$28625401$	$887582698$	$27512735071$	$852889415761$	$26439619335658$	$819628309272001$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.ac $\times$ 1.31.b 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.ac : \(\Q(\sqrt{-30}) \).
• 1.31.b : \(\Q(\sqrt{-123}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.31.ad_cm	$2$	(not in LMFDB)
2.31.b_ci	$2$	(not in LMFDB)
2.31.d_cm	$2$	(not in LMFDB)