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

• Label: 2.31.i_bg
• 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:	$1 + 8 x + 32 x^{2} + 248 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.419615200722$, $\pm0.919615200722$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{46})\)
Galois group:	$C_2^2$
Jacobians:	$26$
Cyclic group of points:	no
Non-cyclic primes:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1250$	$922500$	$902161250$	$851006250000$	$819429281281250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$962$	$30280$	$921478$	$28622200$	$887503682$	$27512874520$	$852892642558$	$26439607667560$	$819628286980802$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{4}}$.

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{46})\).

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

The base change of $A$ to $\F_{31^{4}}$ is 1.923521.abni^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-46}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{31^{2}}$

  The base change of $A$ to $\F_{31^{2}}$ is the simple isogeny class 2.961.a_abni and its endomorphism algebra is \(\Q(i, \sqrt{46})\).

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.ai_bg	$2$	(not in LMFDB)
2.31.a_abe	$8$	(not in LMFDB)
2.31.a_be	$8$	(not in LMFDB)