Abelian variety isogeny class 2.113.aw_ji over $\F_{113}$

• Label: 2.113.aw_ji
• Base field: $\F_{113}$
• 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:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$1 - 22 x + 242 x^{2} - 2486 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0112770403516$, $\pm0.488722959648$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{105})\)
Galois group:	$C_2^2$
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10504$	$163022080$	$2078874885256$	$26576198567526400$	$339450934228729425544$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$92$	$12770$	$1440764$	$162996798$	$18424036732$	$2081951752610$	$235260525836764$	$26584441302894718$	$3004041933605912732$	$339456738992222314850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=38 x^6+100 x^5+57 x^4+104 x^3+40 x^2+73 x+5$
• $y^2=60 x^6+62 x^5+68 x^4+102 x^3+29 x^2+64 x+33$
• $y^2=5 x^6+84 x^5+83 x^4+19 x^3+42 x^2+86 x+28$
• $y^2=31 x^6+23 x^5+85 x^4+71 x^3+82 x^2+17 x+5$
• $y^2=81 x^6+100 x^5+96 x^4+72 x^3+41 x^2+95 x+63$
• $y^2=38 x^6+102 x^5+20 x^4+40 x^3+109 x^2+61 x+68$
• $y^2=74 x^6+70 x^5+90 x^4+90 x^2+43 x+74$
• $y^2=3 x^6+61 x^5+38 x^4+2 x^3+12 x^2+24 x+9$
• $y^2=21 x^6+21 x^5+x^4+10 x^3+71 x^2+17 x+59$
• $y^2=93 x^6+89 x^5+2 x^4+76 x^3+108 x^2+89 x+44$
• $y^2=13 x^6+94 x^5+77 x^4+107 x^3+112 x^2+92 x+40$
• $y^2=70 x^6+37 x^5+59 x^4+61 x^3+62 x^2+87 x+88$
• $y^2=67 x^6+48 x^5+89 x^4+90 x^3+25 x^2+14 x+6$
• $y^2=55 x^6+25 x^5+14 x^4+42 x^3+46 x^2+75 x+73$
• $y^2=50 x^6+22 x^5+90 x^4+35 x^3+22 x^2+14 x+107$
• $y^2=23 x^6+25 x^5+50 x^4+58 x^3+18 x^2+60 x+16$
• $y^2=54 x^6+58 x^5+83 x^4+43 x^3+57 x^2+76 x+25$
• $y^2=x^6+108 x^5+92 x^4+87 x^3+18 x^2+27 x+85$
• $y^2=89 x^6+70 x^5+42 x^4+12 x^3+37 x^2+5 x+5$
• $y^2=45 x^6+22 x^5+80 x^4+95 x^3+5 x^2+46 x+46$
• and

• $y^2=85 x^6+45 x^5+63 x^4+25 x^3+x^2+73 x+95$
• $y^2=43 x^6+105 x^5+19 x^4+19 x^2+8 x+43$
• $y^2=38 x^6+62 x^5+88 x^4+77 x^3+101 x^2+98 x+2$
• $y^2=43 x^6+31 x^5+35 x^4+35 x^2+82 x+43$
• $y^2=36 x^6+5 x^5+89 x^4+24 x^3+9 x^2+38 x+22$
• $y^2=77 x^6+5 x^5+34 x^4+11 x^3+98 x^2+3 x+91$
• $y^2=47 x^6+32 x^5+12 x^4+8 x^3+2 x^2+40 x$
• $y^2=56 x^6+107 x^5+99 x^4+72 x^3+7 x^2+10 x+7$
• $y^2=95 x^6+72 x^5+46 x^4+43 x^3+44 x^2+7 x+103$
• $y^2=81 x^6+62 x^5+35 x^4+50 x^3+49 x^2+98 x+85$
• $y^2=104 x^6+95 x^5+66 x^4+42 x^3+71 x^2+57 x+111$
• $y^2=6 x^6+104 x^5+91 x^4+91 x^2+9 x+6$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113^{4}}$.

Endomorphism algebra over $\F_{113}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{105})\).

Endomorphism algebra over $\overline{\F}_{113}$

The base change of $A$ to $\F_{113^{4}}$ is 1.163047361.ablkk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-105}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{113^{2}}$

  The base change of $A$ to $\F_{113^{2}}$ is the simple isogeny class 2.12769.a_ablkk and its endomorphism algebra is \(\Q(i, \sqrt{105})\).

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
2.113.w_ji	$2$	(not in LMFDB)
2.113.a_aq	$8$	(not in LMFDB)
2.113.a_q	$8$	(not in LMFDB)