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

• Label: 2.83.a_afp
• 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:	$1 - 145 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.0809243439274$, $\pm0.919075656073$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-21}, \sqrt{311})\)
Galois group:	$C_2^2$
Jacobians:	$36$
Isomorphism classes:	48
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6745$	$45495025$	$326940321460$	$2251604518655625$	$15516041193711118225$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6600$	$571788$	$47443828$	$3939040644$	$326940269550$	$27136050989628$	$2252292316934308$	$186940255267540404$	$15516041200216383000$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-21}, \sqrt{311})\).

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.afp^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6531}) \)$)$

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_fp	$4$	(not in LMFDB)