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

• Label: 2.83.a_afl
• 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:	$1 - 141 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.0884827546504$, $\pm0.911517245350$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{307})\)
Galois group:	$C_2^2$
Jacobians:	$27$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6749$	$45549001$	$326940484196$	$2251713088023961$	$15516041194573847789$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6608$	$571788$	$47446116$	$3939040644$	$326940595022$	$27136050989628$	$2252292347479108$	$186940255267540404$	$15516041201941842128$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):

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

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 \(\Q(i, \sqrt{307})\).

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.afl^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-307}) \)$)$

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.ak_hj	$4$	(not in LMFDB)
2.83.a_fl	$4$	(not in LMFDB)
2.83.k_hj	$4$	(not in LMFDB)