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

• Label: 2.83.a_aen
• 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:	$1 - 117 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.125514040080$, $\pm0.874485959920$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{283})\)
Galois group:	$C_2^2$
Jacobians:	$63$
Isomorphism classes:	39
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6773$	$45873529$	$326941189796$	$2252300774644921$	$15516041192685815093$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6656$	$571788$	$47458500$	$3939040644$	$326942006222$	$27136050989628$	$2252292421956484$	$186940255267540404$	$15516041198165776736$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 63 curves (of which all are hyperelliptic):

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

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

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 \(\Q(i, \sqrt{283})\).

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.aen^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-283}) \)$)$

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.ao_ih	$4$	(not in LMFDB)
2.83.a_en	$4$	(not in LMFDB)
2.83.o_ih	$4$	(not in LMFDB)