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

• Label: 2.83.an_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:	$\pm0.0862097840786$, $\pm0.580456882588$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-163})\)
Galois group:	$C_2^2$
Jacobians:	$43$
Isomorphism classes:	30
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5884$	$47472112$	$325753279504$	$2251638920630464$	$15516406250325603124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$6893$	$569708$	$47444553$	$3939133321$	$326940497318$	$27136044103099$	$2252292326807761$	$186940256585763284$	$15516041187916798493$

Jacobians and polarizations

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

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

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

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.aboa^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.n_di	$2$	(not in LMFDB)
2.83.ba_mx	$3$	(not in LMFDB)
2.83.aba_mx	$6$	(not in LMFDB)