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

• Label: 2.31.a_aby
• 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:	$1 - 50 x^{2} + 961 x^{4}$
Frobenius angles:	$\pm0.100692558008$, $\pm0.899307441992$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Galois group:	$C_2^2$
Jacobians:	$26$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$912$	$831744$	$887522832$	$851825627136$	$819628344225552$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$862$	$29792$	$922366$	$28629152$	$887541982$	$27512614112$	$852894063358$	$26439622160672$	$819628401470302$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{7})\).

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

The base change of $A$ to $\F_{31^{2}}$ is 1.961.aby^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

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.ag_br	$3$	(not in LMFDB)
2.31.g_br	$3$	(not in LMFDB)
2.31.a_by	$4$	(not in LMFDB)