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

• Label: 2.83.n_di
• 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:	$1 + 13 x + 86 x^{2} + 1079 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.419543117412$, $\pm0.913790215921$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-163})\)
Galois group:	$C_2^2$
Jacobians:	$43$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8068$	$47472112$	$328131917584$	$2251638920630464$	$15515676133386075148$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$97$	$6893$	$573868$	$47444553$	$3938947967$	$326940497318$	$27136057876157$	$2252292326807761$	$186940253949317524$	$15516041187916798493$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 43 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83^{3}}$.

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-163})\).

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

The base change of $A$ to $\F_{83^{3}}$ is 1.571787.boa^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-163}) \)$)$

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
2.83.an_di	$2$	(not in LMFDB)
2.83.aba_mx	$3$	(not in LMFDB)
2.83.a_ad	$6$	(not in LMFDB)