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

• Label: 2.31.ab_abe
• Base field: $\F_{31}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$1 - x - 30 x^{2} - 31 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.138043034145$, $\pm0.804709700812$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$28$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$900$	$867600$	$882090000$	$854554766400$	$819495125797500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$31$	$901$	$29608$	$925321$	$28624501$	$887605918$	$27512809411$	$852892426801$	$26439637047928$	$819628251354301$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{3}}$.

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{41})\).

Endomorphism algebra over $\overline{\F}_{31}$

The base change of $A$ to $\F_{31^{3}}$ is 1.29791.ado^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

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.b_abe	$2$	(not in LMFDB)
2.31.c_cl	$3$	(not in LMFDB)
2.31.ac_cl	$6$	(not in LMFDB)