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

• Label: 2.83.a_ae
• 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:	$1 - 4 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.246164569905$, $\pm0.753835430095$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-85})\)
Galois group:	$C_2^2$
Jacobians:	$128$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6886$	$47416996$	$326940455974$	$2253598759303056$	$15516041186258890486$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6882$	$571788$	$47485846$	$3939040644$	$326940538578$	$27136050989628$	$2252292043187038$	$186940255267540404$	$15516041185311927522$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 128 curves (of which all are hyperelliptic):

• $y^2=68 x^6+77 x^5+19 x^4+59 x^3+25 x^2+48 x+18$
• $y^2=53 x^6+71 x^5+38 x^4+35 x^3+50 x^2+13 x+36$
• $y^2=4 x^6+81 x^5+24 x^4+42 x^3+16 x^2+12 x+31$
• $y^2=8 x^6+79 x^5+48 x^4+x^3+32 x^2+24 x+62$
• $y^2=60 x^6+67 x^5+14 x^4+60 x^3+39 x^2+35 x+30$
• $y^2=37 x^6+51 x^5+28 x^4+37 x^3+78 x^2+70 x+60$
• $y^2=7 x^6+22 x^5+50 x^4+66 x^3+40 x^2+61 x+14$
• $y^2=14 x^6+44 x^5+17 x^4+49 x^3+80 x^2+39 x+28$
• $y^2=60 x^6+75 x^5+2 x^4+57 x^3+6 x+20$
• $y^2=37 x^6+67 x^5+4 x^4+31 x^3+12 x+40$
• $y^2=8 x^6+63 x^5+71 x^4+24 x^3+14 x^2+x+75$
• $y^2=16 x^6+43 x^5+59 x^4+48 x^3+28 x^2+2 x+67$
• $y^2=26 x^6+14 x^5+63 x^4+45 x^3+2 x^2+23 x+21$
• $y^2=52 x^6+28 x^5+43 x^4+7 x^3+4 x^2+46 x+42$
• $y^2=55 x^6+78 x^5+74 x^4+9 x^3+47 x^2+62 x+24$
• $y^2=27 x^6+73 x^5+65 x^4+18 x^3+11 x^2+41 x+48$
• $y^2=5 x^6+79 x^5+15 x^4+68 x^3+52 x^2+82 x+37$
• $y^2=10 x^6+75 x^5+30 x^4+53 x^3+21 x^2+81 x+74$
• $y^2=7 x^6+3 x^5+6 x^4+28 x^3+33 x^2+60 x+34$
• $y^2=14 x^6+6 x^5+12 x^4+56 x^3+66 x^2+37 x+68$
• and

• $y^2=22 x^6+80 x^5+6 x^4+9 x^3+4 x^2+68 x+61$
• $y^2=44 x^6+77 x^5+12 x^4+18 x^3+8 x^2+53 x+39$
• $y^2=55 x^6+73 x^5+35 x^4+62 x^3+76 x^2+23 x+55$
• $y^2=27 x^6+63 x^5+70 x^4+41 x^3+69 x^2+46 x+27$
• $y^2=29 x^6+58 x^5+10 x^4+x^3+73 x^2+62 x+63$
• $y^2=58 x^6+33 x^5+20 x^4+2 x^3+63 x^2+41 x+43$
• $y^2=64 x^6+31 x^5+18 x^4+50 x^3+11 x^2+41 x+33$
• $y^2=45 x^6+62 x^5+36 x^4+17 x^3+22 x^2+82 x+66$
• $y^2=45 x^6+27 x^5+14 x^4+47 x^3+82 x^2+4 x+1$
• $y^2=7 x^6+54 x^5+28 x^4+11 x^3+81 x^2+8 x+2$
• $y^2=9 x^6+33 x^5+67 x^4+74 x^3+x^2+11 x+46$
• $y^2=18 x^6+66 x^5+51 x^4+65 x^3+2 x^2+22 x+9$
• $y^2=4 x^6+16 x^5+58 x^4+32 x^3+62 x^2+26 x+24$
• $y^2=8 x^6+32 x^5+33 x^4+64 x^3+41 x^2+52 x+48$
• $y^2=35 x^6+29 x^5+17 x^4+67 x^3+21 x^2+42 x+60$
• $y^2=70 x^6+58 x^5+34 x^4+51 x^3+42 x^2+x+37$
• $y^2=65 x^6+49 x^5+69 x^4+52 x^3+71 x^2+9$
• $y^2=47 x^6+15 x^5+55 x^4+21 x^3+59 x^2+18$
• $y^2=26 x^6+47 x^5+44 x^4+15 x^3+55 x^2+13 x+11$
• $y^2=52 x^6+11 x^5+5 x^4+30 x^3+27 x^2+26 x+22$
• $y^2=32 x^6+75 x^5+56 x^4+22 x^3+34 x^2+78$
• $y^2=64 x^6+67 x^5+29 x^4+44 x^3+68 x^2+73$
• $y^2=11 x^6+17 x^5+27 x^4+12 x^3+28 x^2+18 x+3$
• $y^2=22 x^6+34 x^5+54 x^4+24 x^3+56 x^2+36 x+6$
• $y^2=19 x^6+73 x^5+79 x^3+4 x^2+10 x+24$
• $y^2=38 x^6+63 x^5+75 x^3+8 x^2+20 x+48$
• $y^2=31 x^6+31 x^5+33 x^4+67 x^3+69 x^2+26 x+80$
• $y^2=62 x^6+62 x^5+66 x^4+51 x^3+55 x^2+52 x+77$
• $y^2=56 x^6+72 x^5+57 x^4+33 x^3+81 x^2+75 x+50$
• $y^2=29 x^6+61 x^5+31 x^4+66 x^3+79 x^2+67 x+17$
• $y^2=19 x^6+x^5+34 x^4+16 x^3+65 x^2+62 x+15$
• $y^2=38 x^6+2 x^5+68 x^4+32 x^3+47 x^2+41 x+30$
• $y^2=65 x^6+22 x^5+22 x^4+37 x^3+7 x^2+36 x+47$
• $y^2=47 x^6+44 x^5+44 x^4+74 x^3+14 x^2+72 x+11$
• $y^2=x^6+68 x^5+12 x^4+68 x^3+x^2+x+18$
• $y^2=2 x^6+53 x^5+24 x^4+53 x^3+2 x^2+2 x+36$
• $y^2=24 x^6+31 x^5+42 x^4+61 x^3+77 x^2+69 x+7$
• $y^2=48 x^6+62 x^5+x^4+39 x^3+71 x^2+55 x+14$
• $y^2=44 x^6+17 x^5+64 x^4+31 x^3+71 x^2+24 x+11$
• $y^2=5 x^6+34 x^5+45 x^4+62 x^3+59 x^2+48 x+22$
• $y^2=73 x^6+34 x^5+77 x^4+37 x^3+8 x^2+36 x+20$
• $y^2=63 x^6+68 x^5+71 x^4+74 x^3+16 x^2+72 x+40$
• $y^2=73 x^6+2 x^5+74 x^4+48 x^3+35 x^2+63 x+29$
• $y^2=63 x^6+4 x^5+65 x^4+13 x^3+70 x^2+43 x+58$
• $y^2=35 x^6+71 x^5+70 x^4+48 x^3+31 x^2+36 x+8$
• $y^2=70 x^6+59 x^5+57 x^4+13 x^3+62 x^2+72 x+16$
• $y^2=60 x^6+22 x^5+45 x^4+41 x^3+6 x^2+16 x+72$
• $y^2=37 x^6+44 x^5+7 x^4+82 x^3+12 x^2+32 x+61$
• $y^2=48 x^6+38 x^5+61 x^4+26 x^3+10 x^2+23 x+6$
• $y^2=13 x^6+76 x^5+39 x^4+52 x^3+20 x^2+46 x+12$
• $y^2=8 x^6+21 x^5+15 x^4+69 x^3+42 x^2+45 x+44$
• $y^2=16 x^6+42 x^5+30 x^4+55 x^3+x^2+7 x+5$
• $y^2=38 x^6+62 x^5+56 x^4+71 x^3+78 x^2+47 x+31$
• $y^2=76 x^6+41 x^5+29 x^4+59 x^3+73 x^2+11 x+62$
• $y^2=64 x^6+55 x^5+25 x^4+77 x^3+8 x^2+67 x+5$
• $y^2=45 x^6+27 x^5+50 x^4+71 x^3+16 x^2+51 x+10$
• $y^2=16 x^6+26 x^4+78 x^3+21 x^2+29 x+55$
• $y^2=32 x^6+52 x^4+73 x^3+42 x^2+58 x+27$
• $y^2=41 x^6+23 x^5+29 x^4+66 x^3+60 x^2+55 x+18$
• $y^2=82 x^6+46 x^5+58 x^4+49 x^3+37 x^2+27 x+36$
• $y^2=16 x^6+4 x^5+69 x^4+67 x^3+6 x^2+35 x+74$
• $y^2=32 x^6+8 x^5+55 x^4+51 x^3+12 x^2+70 x+65$
• $y^2=59 x^6+32 x^5+59 x^4+61 x^3+72 x^2+10 x+31$
• $y^2=35 x^6+64 x^5+35 x^4+39 x^3+61 x^2+20 x+62$
• $y^2=48 x^6+22 x^5+77 x^4+7 x^3+14 x^2+74 x+10$
• $y^2=13 x^6+44 x^5+71 x^4+14 x^3+28 x^2+65 x+20$
• $y^2=34 x^6+72 x^5+64 x^4+77 x^3+40 x^2+x+81$
• $y^2=68 x^6+61 x^5+45 x^4+71 x^3+80 x^2+2 x+79$
• $y^2=26 x^6+42 x^5+3 x^4+17 x^3+77 x^2+71 x+23$
• $y^2=52 x^6+x^5+6 x^4+34 x^3+71 x^2+59 x+46$
• $y^2=25 x^6+67 x^5+10 x^4+27 x^3+30 x^2+69 x+19$
• $y^2=50 x^6+51 x^5+20 x^4+54 x^3+60 x^2+55 x+38$
• $y^2=42 x^6+24 x^5+4 x^4+66 x^3+61 x^2+12 x+49$
• $y^2=x^6+48 x^5+8 x^4+49 x^3+39 x^2+24 x+15$
• $y^2=62 x^6+77 x^5+18 x^4+53 x^3+60 x^2+11 x+58$
• $y^2=41 x^6+71 x^5+36 x^4+23 x^3+37 x^2+22 x+33$
• $y^2=39 x^6+35 x^5+40 x^4+17 x^3+45 x^2+73 x+81$
• $y^2=78 x^6+70 x^5+80 x^4+34 x^3+7 x^2+63 x+79$
• $y^2=40 x^6+44 x^5+29 x^4+3 x^3+26 x^2+33 x+63$
• $y^2=80 x^6+5 x^5+58 x^4+6 x^3+52 x^2+66 x+43$
• $y^2=22 x^6+58 x^5+6 x^4+50 x^3+12 x^2+33 x+25$
• $y^2=44 x^6+33 x^5+12 x^4+17 x^3+24 x^2+66 x+50$
• $y^2=77 x^6+71 x^5+61 x^4+82 x^3+21 x^2+4 x+55$
• $y^2=71 x^6+59 x^5+39 x^4+81 x^3+42 x^2+8 x+27$
• $y^2=17 x^6+3 x^5+45 x^4+80 x^3+9 x^2+53 x+35$
• $y^2=34 x^6+6 x^5+7 x^4+77 x^3+18 x^2+23 x+70$
• $y^2=54 x^6+40 x^5+41 x^4+70 x^3+31 x^2+10 x+29$
• $y^2=25 x^6+80 x^5+82 x^4+57 x^3+62 x^2+20 x+58$
• $y^2=37 x^6+71 x^5+31 x^4+33 x^3+82 x^2+14 x+75$
• $y^2=74 x^6+59 x^5+62 x^4+66 x^3+81 x^2+28 x+67$
• $y^2=50 x^6+45 x^5+28 x^4+26 x^3+12 x^2+27 x+48$
• $y^2=17 x^6+7 x^5+56 x^4+52 x^3+24 x^2+54 x+13$
• $y^2=73 x^6+71 x^5+10 x^4+13 x^3+39 x^2+70 x+25$
• $y^2=63 x^6+59 x^5+20 x^4+26 x^3+78 x^2+57 x+50$
• $y^2=65 x^6+48 x^5+13 x^4+3 x^3+63 x^2+56 x+10$
• $y^2=47 x^6+13 x^5+26 x^4+6 x^3+43 x^2+29 x+20$
• $y^2=72 x^6+53 x^5+64 x^4+57 x^3+78 x^2+66 x+6$
• $y^2=61 x^6+23 x^5+45 x^4+31 x^3+73 x^2+49 x+12$
• $y^2=57 x^6+70 x^5+50 x^4+34 x^3+13 x^2+72 x+81$
• $y^2=31 x^6+57 x^5+17 x^4+68 x^3+26 x^2+61 x+79$
• $y^2=15 x^6+68 x^5+14 x^4+62 x^3+5 x^2+12 x+7$
• $y^2=30 x^6+53 x^5+28 x^4+41 x^3+10 x^2+24 x+14$
• $y^2=7 x^6+60 x^5+51 x^4+34 x^3+16 x^2+31 x+60$
• $y^2=14 x^6+37 x^5+19 x^4+68 x^3+32 x^2+62 x+37$
• $y^2=31 x^6+61 x^5+48 x^4+43 x^3+10 x^2+13 x+60$
• $y^2=62 x^6+39 x^5+13 x^4+3 x^3+20 x^2+26 x+37$
• $y^2=60 x^6+72 x^5+67 x^4+74 x^3+6 x^2+43 x+82$
• $y^2=37 x^6+61 x^5+51 x^4+65 x^3+12 x^2+3 x+81$

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 \(\Q(\sqrt{-2}, \sqrt{-85})\).

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-85}) \)$)$

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_e	$4$	(not in LMFDB)
2.83.as_gg	$8$	(not in LMFDB)
2.83.s_gg	$8$	(not in LMFDB)