Abelian variety isogeny class 2.83.a_bz over $\F_{83}$

• Label: 2.83.a_bz
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 + 51 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.299700864398$, $\pm0.700299135602$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{115}, \sqrt{-217})\)
Galois group:	$C_2^2$
Jacobians:	$84$
Isomorphism classes:	192

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6941$	$48177481$	$326939452004$	$2253353335311001$	$15516041195083586861$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6992$	$571788$	$47480676$	$3939040644$	$326938530638$	$27136050989628$	$2252292172121668$	$186940255267540404$	$15516041202961320272$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 84 curves (of which all are hyperelliptic):

• $y^2=51 x^6+57 x^5+20 x^4+42 x^3+59 x^2+43 x+58$
• $y^2=18 x^6+11 x^5+59 x^4+49 x^3+67 x^2+58 x+35$
• $y^2=21 x^6+81 x^5+23 x^3+80 x^2+68 x+11$
• $y^2=42 x^6+79 x^5+46 x^3+77 x^2+53 x+22$
• $y^2=66 x^6+7 x^5+12 x^4+58 x^3+27 x^2+9 x+27$
• $y^2=49 x^6+14 x^5+24 x^4+33 x^3+54 x^2+18 x+54$
• $y^2=71 x^6+34 x^5+21 x^4+74 x^3+19 x^2+64 x+22$
• $y^2=59 x^6+68 x^5+42 x^4+65 x^3+38 x^2+45 x+44$
• $y^2=44 x^6+17 x^5+28 x^4+59 x^3+3 x^2+58 x+76$
• $y^2=36 x^6+20 x^5+45 x^4+39 x^3+23 x^2+24 x+47$
• $y^2=72 x^6+40 x^5+7 x^4+78 x^3+46 x^2+48 x+11$
• $y^2=73 x^6+5 x^5+7 x^4+41 x^3+16 x^2+50 x+35$
• $y^2=63 x^6+10 x^5+14 x^4+82 x^3+32 x^2+17 x+70$
• $y^2=30 x^6+7 x^5+39 x^4+68 x^3+46 x^2+18 x+75$
• $y^2=60 x^6+14 x^5+78 x^4+53 x^3+9 x^2+36 x+67$
• $y^2=31 x^6+55 x^5+30 x^4+43 x^3+16 x^2+15 x+38$
• $y^2=62 x^6+27 x^5+60 x^4+3 x^3+32 x^2+30 x+76$
• $y^2=18 x^6+63 x^5+61 x^4+22 x^3+24 x^2+26 x+69$
• $y^2=36 x^6+43 x^5+39 x^4+44 x^3+48 x^2+52 x+55$
• $y^2=17 x^6+25 x^5+39 x^4+5 x^3+9 x^2+36 x+37$
• and

• $y^2=34 x^6+50 x^5+78 x^4+10 x^3+18 x^2+72 x+74$
• $y^2=63 x^6+58 x^5+54 x^4+82 x^3+40 x^2+3 x+52$
• $y^2=36 x^6+74 x^5+25 x^4+67 x^3+27 x^2+23 x+40$
• $y^2=72 x^6+65 x^5+50 x^4+51 x^3+54 x^2+46 x+80$
• $y^2=67 x^6+21 x^5+56 x^4+37 x^3+24 x^2+72 x+53$
• $y^2=51 x^6+42 x^5+29 x^4+74 x^3+48 x^2+61 x+23$
• $y^2=74 x^6+39 x^5+76 x^4+41 x^3+33 x^2+8 x+77$
• $y^2=65 x^6+78 x^5+69 x^4+82 x^3+66 x^2+16 x+71$
• $y^2=51 x^6+20 x^5+47 x^4+37 x^3+38 x^2+26 x+17$
• $y^2=67 x^6+71 x^5+72 x^4+5 x^3+67 x^2+82 x+29$
• $y^2=72 x^6+36 x^5+44 x^4+27 x^3+33 x^2+13 x+46$
• $y^2=74 x^6+10 x^5+69 x^4+53 x^3+14 x^2+52 x+59$
• $y^2=65 x^6+20 x^5+55 x^4+23 x^3+28 x^2+21 x+35$
• $y^2=49 x^6+22 x^5+70 x^4+38 x^3+80 x^2+21 x+82$
• $y^2=27 x^6+27 x^5+82 x^4+53 x^3+25 x^2+47 x+75$
• $y^2=54 x^6+54 x^5+81 x^4+23 x^3+50 x^2+11 x+67$
• $y^2=34 x^6+15 x^5+30 x^3+79 x^2+53 x+11$
• $y^2=68 x^6+30 x^5+60 x^3+75 x^2+23 x+22$
• $y^2=13 x^6+82 x^5+55 x^4+4 x^3+37 x^2+3 x+19$
• $y^2=26 x^6+81 x^5+27 x^4+8 x^3+74 x^2+6 x+38$
• $y^2=69 x^6+66 x^5+30 x^4+53 x^3+14 x^2+9 x+68$
• $y^2=55 x^6+49 x^5+60 x^4+23 x^3+28 x^2+18 x+53$
• $y^2=13 x^6+39 x^5+40 x^4+42 x^3+44 x^2+26 x+41$
• $y^2=26 x^6+78 x^5+80 x^4+x^3+5 x^2+52 x+82$
• $y^2=14 x^6+9 x^5+30 x^4+25 x^3+74 x^2+48 x+51$
• $y^2=28 x^6+18 x^5+60 x^4+50 x^3+65 x^2+13 x+19$
• $y^2=46 x^6+4 x^5+66 x^4+18 x^3+37 x^2+35 x+28$
• $y^2=9 x^6+8 x^5+49 x^4+36 x^3+74 x^2+70 x+56$
• $y^2=34 x^6+39 x^5+49 x^4+54 x^2+12 x+78$
• $y^2=68 x^6+78 x^5+15 x^4+25 x^2+24 x+73$
• $y^2=34 x^6+30 x^5+42 x^4+30 x^3+4 x^2+22 x+46$
• $y^2=68 x^6+60 x^5+x^4+60 x^3+8 x^2+44 x+9$
• $y^2=66 x^6+11 x^5+5 x^4+42 x^3+53 x^2+22 x+68$
• $y^2=49 x^6+22 x^5+10 x^4+x^3+23 x^2+44 x+53$
• $y^2=25 x^6+34 x^5+54 x^4+29 x^3+16 x^2+51 x+58$
• $y^2=50 x^6+68 x^5+25 x^4+58 x^3+32 x^2+19 x+33$
• $y^2=77 x^6+26 x^5+9 x^4+64 x^3+41 x^2+2 x+19$
• $y^2=71 x^6+52 x^5+18 x^4+45 x^3+82 x^2+4 x+38$
• $y^2=16 x^6+4 x^5+30 x^4+2 x^3+25 x^2+39 x+17$
• $y^2=12 x^6+25 x^5+12 x^4+54 x^3+44 x^2+52 x+41$
• $y^2=24 x^6+50 x^5+24 x^4+25 x^3+5 x^2+21 x+82$
• $y^2=78 x^6+74 x^5+8 x^4+67 x^3+81 x^2+66 x+57$
• $y^2=73 x^6+65 x^5+16 x^4+51 x^3+79 x^2+49 x+31$
• $y^2=41 x^6+20 x^5+76 x^4+22 x^3+23 x^2+28 x+8$
• $y^2=39 x^6+77 x^5+15 x^4+67 x^3+70 x^2+2 x+33$
• $y^2=15 x^6+64 x^5+70 x^4+82 x^3+26 x^2+58 x+40$
• $y^2=30 x^6+45 x^5+57 x^4+81 x^3+52 x^2+33 x+80$
• $y^2=27 x^6+8 x^5+21 x^4+81 x^3+70 x^2+61 x+54$
• $y^2=82 x^6+64 x^5+58 x^4+2 x^3+64 x^2+79 x+41$
• $y^2=81 x^6+45 x^5+33 x^4+4 x^3+45 x^2+75 x+82$
• $y^2=21 x^6+52 x^5+74 x^4+82 x^3+66 x^2+33 x+55$
• $y^2=42 x^6+21 x^5+65 x^4+81 x^3+49 x^2+66 x+27$
• $y^2=58 x^6+18 x^5+63 x^4+40 x^3+33 x^2+46 x+38$
• $y^2=33 x^6+36 x^5+43 x^4+80 x^3+66 x^2+9 x+76$
• $y^2=27 x^6+49 x^5+29 x^4+80 x^3+32 x^2+11 x+61$
• $y^2=54 x^6+15 x^5+58 x^4+77 x^3+64 x^2+22 x+39$
• $y^2=32 x^6+25 x^5+x^4+31 x^3+26 x^2+49 x+20$
• $y^2=64 x^6+50 x^5+2 x^4+62 x^3+52 x^2+15 x+40$
• $y^2=15 x^6+19 x^5+65 x^4+27 x^3+50 x^2+34 x+30$
• $y^2=30 x^6+38 x^5+47 x^4+54 x^3+17 x^2+68 x+60$
• $y^2=39 x^6+9 x^5+3 x^4+67 x^3+76 x^2+71 x+8$
• $y^2=78 x^6+18 x^5+6 x^4+51 x^3+69 x^2+59 x+16$
• $y^2=67 x^6+x^5+58 x^4+21 x^3+35 x^2+28 x+58$
• $y^2=51 x^6+2 x^5+33 x^4+42 x^3+70 x^2+56 x+33$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83^{2}}$.

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{115}, \sqrt{-217})\).

Endomorphism algebra over $\overline{\F}_{83}$

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.bz^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-24955}) \)$)$

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
2.83.a_abz	$4$	(not in LMFDB)