Abelian variety isogeny class 2.97.ar_im over $\F_{97}$

• Label: 2.97.ar_im
• Base field: $\F_{97}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 17 x + 220 x^{2} - 1649 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.216853147274$, $\pm0.472506156356$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-1078 +34 \sqrt{185}})\)
Galois group:	$D_{4}$
Jacobians:	$98$
Isomorphism classes:	196
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7964$	$89961344$	$834215493056$	$7837389467680256$	$73743282616120852284$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$9561$	$914034$	$88528785$	$8587441561$	$832974658302$	$80798293682985$	$7837433338842849$	$760231055707664178$	$73742412685826268361$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 98 curves (of which all are hyperelliptic):

• $y^2=39 x^6+68 x^5+41 x^4+84 x^3+36 x^2+65 x+64$
• $y^2=7 x^6+32 x^5+87 x^4+66 x^3+53 x^2+71 x+52$
• $y^2=29 x^6+34 x^5+40 x^4+40 x^3+95 x^2+79 x+81$
• $y^2=20 x^6+10 x^5+52 x^4+57 x^3+18 x^2+56 x+1$
• $y^2=90 x^6+20 x^5+41 x^4+83 x^3+32 x^2+61 x+75$
• $y^2=56 x^6+95 x^5+82 x^4+37 x^3+32 x^2+35 x+71$
• $y^2=84 x^6+57 x^5+61 x^4+24 x^3+30 x^2+67 x+82$
• $y^2=26 x^6+87 x^5+8 x^4+13 x^3+85 x^2+64 x+60$
• $y^2=63 x^6+79 x^5+4 x^4+60 x^3+22 x^2+31 x+47$
• $y^2=75 x^6+42 x^5+9 x^4+47 x^3+80 x^2+77 x+85$
• $y^2=38 x^6+32 x^5+49 x^4+85 x^3+62 x^2+69 x+13$
• $y^2=78 x^6+85 x^5+45 x^4+82 x^3+15 x^2+92 x+41$
• $y^2=53 x^6+82 x^5+35 x^3+46 x^2+33 x+9$
• $y^2=41 x^6+65 x^5+28 x^4+30 x^3+80 x^2+11 x+86$
• $y^2=41 x^6+37 x^5+84 x^4+62 x^3+82 x^2+49 x+30$
• $y^2=22 x^6+26 x^5+23 x^4+65 x^3+35 x^2+6 x+21$
• $y^2=33 x^6+10 x^5+4 x^4+8 x^3+16 x^2+79 x+76$
• $y^2=17 x^6+x^5+90 x^4+43 x^3+90 x^2+3 x+67$
• $y^2=23 x^6+70 x^5+31 x^4+44 x^3+55 x^2+52 x+78$
• $y^2=42 x^6+72 x^5+2 x^4+35 x^3+61 x^2+20 x+89$
• and

• $y^2=83 x^6+18 x^5+85 x^4+88 x^3+75 x^2+28 x+57$
• $y^2=87 x^6+42 x^5+53 x^4+34 x^3+63 x+57$
• $y^2=51 x^6+75 x^5+37 x^4+40 x^3+56 x^2+73 x+70$
• $y^2=92 x^6+50 x^5+21 x^4+55 x^3+17 x^2+34 x+63$
• $y^2=82 x^6+61 x^5+15 x^4+78 x^3+66 x^2+96 x+42$
• $y^2=4 x^6+91 x^5+13 x^4+21 x^3+89 x^2+12 x+76$
• $y^2=68 x^6+70 x^5+59 x^4+20 x^3+89 x^2+90 x+72$
• $y^2=10 x^6+27 x^5+68 x^4+36 x^3+53 x^2+26 x+78$
• $y^2=53 x^6+34 x^5+10 x^4+10 x^3+57 x^2+69 x+10$
• $y^2=28 x^6+49 x^5+47 x^4+2 x^3+27 x+7$
• $y^2=84 x^6+79 x^5+88 x^4+50 x^3+x^2+30 x+26$
• $y^2=91 x^6+78 x^5+32 x^4+80 x^3+44 x^2+95 x+38$
• $y^2=30 x^6+42 x^5+8 x^3+85 x^2+93 x+39$
• $y^2=29 x^6+52 x^5+53 x^4+14 x^3+28 x^2+48 x+65$
• $y^2=55 x^6+36 x^5+79 x^4+25 x^3+92 x^2+96 x+35$
• $y^2=75 x^6+5 x^5+77 x^4+68 x^3+30 x^2+23 x+48$
• $y^2=35 x^6+51 x^5+75 x^4+94 x^3+8 x^2+10 x+75$
• $y^2=75 x^6+4 x^5+96 x^4+88 x^3+16 x^2+84$
• $y^2=75 x^6+69 x^5+96 x^4+37 x^3+34 x^2+6 x+92$
• $y^2=13 x^6+87 x^5+56 x^4+40 x^3+44 x^2+29 x+94$
• $y^2=23 x^6+21 x^5+30 x^4+92 x^3+5 x^2+81 x+92$
• $y^2=37 x^6+93 x^5+15 x^4+73 x^3+14 x^2+50 x+31$
• $y^2=13 x^6+35 x^4+12 x^3+91 x^2+33 x+48$
• $y^2=49 x^6+4 x^5+72 x^4+16 x^3+46 x^2+26 x+20$
• $y^2=4 x^6+40 x^5+27 x^4+42 x^3+48 x^2+88 x+24$
• $y^2=26 x^6+37 x^5+11 x^4+44 x^3+53 x^2+73 x+81$
• $y^2=36 x^6+50 x^5+52 x^4+2 x^3+58 x^2+64 x+4$
• $y^2=65 x^6+71 x^5+78 x^4+19 x^3+67 x^2+21 x+16$
• $y^2=85 x^6+80 x^5+35 x^4+67 x^3+17 x^2+7 x+69$
• $y^2=80 x^6+67 x^5+46 x^4+92 x^3+25 x^2+50 x+24$
• $y^2=83 x^6+36 x^5+30 x^4+59 x^3+74 x^2+x+36$
• $y^2=51 x^6+66 x^5+6 x^4+26 x^3+95 x^2+62 x+69$
• $y^2=24 x^6+15 x^5+81 x^4+39 x^3+71 x^2+62 x+38$
• $y^2=73 x^6+15 x^5+28 x^4+9 x^3+77 x^2+81 x+66$
• $y^2=35 x^6+10 x^5+34 x^4+37 x^3+70 x^2+46 x+46$
• $y^2=63 x^6+41 x^5+19 x^4+66 x^3+8 x^2+39 x+30$
• $y^2=83 x^6+13 x^5+59 x^4+12 x^3+67 x^2+36 x$
• $y^2=68 x^6+2 x^5+65 x^4+89 x^3+73 x^2+85 x+82$
• $y^2=69 x^6+63 x^5+88 x^4+69 x^3+79 x^2+70 x+69$
• $y^2=54 x^6+36 x^5+52 x^4+62 x^3+x^2+25 x+4$
• $y^2=72 x^6+13 x^5+75 x^4+91 x^3+43 x^2+65 x+90$
• $y^2=38 x^6+78 x^5+22 x^4+91 x^3+36 x^2+x+30$
• $y^2=43 x^6+91 x^5+58 x^4+7 x^3+12 x^2+62 x+83$
• $y^2=77 x^6+83 x^5+32 x^4+89 x^3+58 x^2+30$
• $y^2=86 x^6+40 x^5+95 x^4+53 x^3+24 x^2+30 x+47$
• $y^2=22 x^6+22 x^5+7 x^4+68 x^3+x^2+69 x+79$
• $y^2=28 x^6+4 x^5+52 x^4+18 x^3+36 x^2+29 x+55$
• $y^2=19 x^6+96 x^5+46 x^4+26 x^3+11 x^2+50 x+72$
• $y^2=78 x^6+28 x^5+74 x^4+47 x^3+34 x^2+50 x+55$
• $y^2=43 x^6+41 x^5+54 x^4+54 x^3+35 x^2+9 x+69$
• $y^2=74 x^6+76 x^5+45 x^4+31 x^3+32 x^2+74 x+21$
• $y^2=96 x^6+41 x^5+34 x^4+6 x^3+20 x^2+89 x+49$
• $y^2=16 x^6+49 x^5+8 x^4+63 x^3+91 x^2+20 x+85$
• $y^2=65 x^6+70 x^5+11 x^4+16 x^3+95 x^2+75 x+82$
• $y^2=20 x^6+39 x^5+30 x^4+30 x^3+17 x^2+65 x+57$
• $y^2=25 x^6+73 x^5+52 x^4+33 x^3+29 x^2+38 x+23$
• $y^2=76 x^6+66 x^5+86 x^4+13 x^3+12 x^2+62 x+84$
• $y^2=41 x^6+61 x^4+67 x^3+14 x^2+34 x+89$
• $y^2=45 x^6+42 x^5+27 x^4+4 x^3+71 x^2+83 x+46$
• $y^2=53 x^6+84 x^5+96 x^4+63 x^3+68 x^2+40 x+57$
• $y^2=70 x^6+44 x^5+73 x^4+87 x^3+80 x^2+72 x+34$
• $y^2=23 x^6+37 x^5+79 x^4+59 x^3+91 x^2+76 x+38$
• $y^2=94 x^6+14 x^5+51 x^4+73 x^3+66 x^2+63 x+10$
• $y^2=34 x^6+6 x^5+30 x^4+18 x^3+36 x^2+49 x+23$
• $y^2=96 x^6+74 x^5+14 x^4+83 x^3+87 x^2+48 x+13$
• $y^2=2 x^6+42 x^5+81 x^4+51 x^3+17 x+20$
• $y^2=64 x^6+13 x^5+79 x^4+67 x^3+8 x^2+41 x+21$
• $y^2=2 x^6+60 x^5+66 x^4+16 x^3+25 x^2+87 x+41$
• $y^2=7 x^6+x^5+63 x^4+43 x^3+6 x^2+83 x+81$
• $y^2=16 x^6+72 x^5+6 x^4+84 x^3+13 x^2+79 x+53$
• $y^2=32 x^6+91 x^5+81 x^4+33 x^3+93 x^2+41 x+73$
• $y^2=50 x^6+47 x^5+88 x^4+17 x^3+69 x^2+51 x+7$
• $y^2=93 x^6+48 x^5+21 x^4+60 x^3+38 x^2+16 x+40$
• $y^2=84 x^6+63 x^5+54 x^4+44 x^3+59 x^2+2 x+77$
• $y^2=90 x^6+8 x^5+61 x^4+81 x^3+30 x^2+93 x+94$
• $y^2=83 x^6+71 x^5+55 x^4+29 x^3+26 x^2+2 x+64$
• $y^2=13 x^6+58 x^5+52 x^4+95 x^3+34 x^2+88 x+63$
• $y^2=58 x^6+81 x^5+8 x^4+56 x^3+57 x^2+63 x+57$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1078 +34 \sqrt{185}})\).

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
2.97.r_im	$2$	(not in LMFDB)