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

• Label: 2.83.ap_im
• Base field: $\F_{83}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 83 x^{2} )( 1 - 6 x + 83 x^{2} )$
	$1 - 15 x + 220 x^{2} - 1245 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.335556542753$, $\pm0.393189690303$
Angle rank:	$2$ (numerical)
Jacobians:	$70$
Cyclic group of points:	no
Non-cyclic primes:	$3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5850$	$48964500$	$328538737800$	$2252455136100000$	$15515287566434691750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$7105$	$574578$	$47461753$	$3938849319$	$326938741090$	$27136053648213$	$2252292369283633$	$186940256025669894$	$15516041182960701025$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 70 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.aj $\times$ 1.83.ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.83.aj : \(\Q(\sqrt{-251}) \).
• 1.83.ag : \(\Q(\sqrt{-74}) \).

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.ad_ei	$2$	(not in LMFDB)
2.83.d_ei	$2$	(not in LMFDB)
2.83.p_im	$2$	(not in LMFDB)