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

• Label: 2.31.e_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:	$( 1 + 2 x + 31 x^{2} )^{2}$
	$1 + 4 x + 66 x^{2} + 124 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.557482058976$, $\pm0.557482058976$
Angle rank:	$1$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1156$	$1040400$	$876988996$	$850231526400$	$820109499102916$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1078$	$29436$	$920638$	$28645956$	$887559478$	$27511981596$	$852890572798$	$26439642697956$	$819628260310198$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$

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.ae_co	$2$	(not in LMFDB)
2.31.a_cg	$2$	(not in LMFDB)
2.31.ac_abb	$3$	(not in LMFDB)