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

• Label: 2.31.a_abs
• 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:	$1 - 44 x^{2} + 961 x^{4}$
Frobenius angles:	$\pm0.124420346278$, $\pm0.875579653722$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-53})\)
Galois group:	$C_2^2$
Jacobians:	$18$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$918$	$842724$	$887545350$	$852867026064$	$819628328199078$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$874$	$29792$	$923494$	$28629152$	$887587018$	$27512614112$	$852894731134$	$26439622160672$	$819628369417354$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{2}}$.

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-53})\).

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

The base change of $A$ to $\F_{31^{2}}$ is 1.961.abs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-53}) \)$)$

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.a_bs	$4$	(not in LMFDB)
2.31.ag_s	$8$	(not in LMFDB)
2.31.g_s	$8$	(not in LMFDB)