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

• Label: 2.31.ap_ei
• 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:	$( 1 - 10 x + 31 x^{2} )( 1 - 5 x + 31 x^{2} )$
	$1 - 15 x + 112 x^{2} - 465 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.145000771013$, $\pm0.351775594290$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$594$	$923076$	$895583304$	$853845300000$	$819604671152454$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$961$	$30062$	$924553$	$28628327$	$887502346$	$27512906417$	$852894197233$	$26439636990962$	$819628292744281$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.ak $\times$ 1.31.af 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.ak : \(\Q(\sqrt{-6}) \).
• 1.31.af : \(\Q(\sqrt{-11}) \).

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.af_m	$2$	(not in LMFDB)
2.31.f_m	$2$	(not in LMFDB)
2.31.p_ei	$2$	(not in LMFDB)