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

• Label: 2.97.b_hk
• Base field: $\F_{97}$
• 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:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 97 x^{2} )( 1 + 2 x + 97 x^{2} )$
	$1 + x + 192 x^{2} + 97 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.483833314357$, $\pm0.532375263179$
Angle rank:	$2$ (numerical)
Jacobians:	$84$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9700$	$92188800$	$832714464400$	$7834272443712000$	$73742787806075558500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$99$	$9793$	$912390$	$88493569$	$8587383939$	$832975242046$	$80798278990563$	$7837433310124801$	$760231059267432390$	$73742412713530641793$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 84 curves (of which all are hyperelliptic):

• $y^2=51 x^6+90 x^5+30 x^4+87 x^3+46 x^2+17 x+8$
• $y^2=12 x^6+96 x^5+90 x^3+14 x^2+78 x+72$
• $y^2=73 x^6+42 x^5+20 x^4+81 x^3+61 x^2+92 x+29$
• $y^2=93 x^6+44 x^5+29 x^4+60 x^3+32 x^2+5 x+93$
• $y^2=64 x^6+38 x^5+67 x^4+23 x^3+17 x^2+80 x+89$
• $y^2=38 x^6+50 x^5+3 x^4+77 x^3+86 x^2+84 x+12$
• $y^2=53 x^6+84 x^5+88 x^4+16 x^3+18 x^2+39 x+85$
• $y^2=91 x^6+32 x^5+84 x^4+35 x^3+64 x^2+31 x+15$
• $y^2=26 x^6+60 x^5+24 x^4+2 x^3+67 x^2+96 x+41$
• $y^2=95 x^6+93 x^5+96 x^4+20 x^3+40 x^2+63 x+48$
• $y^2=34 x^6+70 x^5+18 x^4+18 x^3+21 x^2+93 x+22$
• $y^2=5 x^6+16 x^5+90 x^4+38 x^3+26 x^2+16 x+8$
• $y^2=13 x^6+19 x^5+82 x^4+9 x^3+81 x^2+21 x+66$
• $y^2=41 x^6+55 x^5+38 x^4+35 x^3+4 x^2+41 x+88$
• $y^2=11 x^6+72 x^5+52 x^4+54 x^3+91 x^2+8 x+3$
• $y^2=41 x^6+46 x^5+77 x^4+96 x^3+69 x^2+23 x+14$
• $y^2=9 x^6+85 x^5+4 x^4+91 x^3+47 x^2+5 x+37$
• $y^2=6 x^6+28 x^5+56 x^4+36 x^3+9 x^2+77 x+52$
• $y^2=48 x^6+47 x^5+96 x^4+x^3+56 x^2+53 x+67$
• $y^2=13 x^6+35 x^5+60 x^4+89 x^3+35 x^2+96 x+51$
• and

• $y^2=43 x^6+35 x^5+43 x^4+76 x^3+25 x^2+33 x+45$
• $y^2=18 x^6+77 x^5+24 x^4+67 x^3+45 x^2+28 x+76$
• $y^2=7 x^6+35 x^5+96 x^4+19 x^3+86 x^2+65 x+15$
• $y^2=88 x^6+65 x^5+74 x^4+70 x^3+39 x^2+89 x+85$
• $y^2=44 x^6+16 x^5+6 x^4+77 x^3+33 x^2+77 x+9$
• $y^2=59 x^6+18 x^5+73 x^4+69 x^3+75 x^2+76 x+67$
• $y^2=54 x^6+95 x^5+41 x^4+10 x^3+10 x^2+52 x+44$
• $y^2=8 x^6+32 x^5+68 x^4+37 x^3+39 x^2+93 x+79$
• $y^2=81 x^6+95 x^5+29 x^4+29 x^3+56 x^2+80 x+71$
• $y^2=50 x^6+2 x^5+12 x^4+34 x^3+32 x^2+4 x+10$
• $y^2=62 x^6+60 x^5+21 x^4+48 x^3+24 x^2+78 x+55$
• $y^2=68 x^6+54 x^5+83 x^4+87 x^3+81 x^2+9 x+5$
• $y^2=21 x^6+91 x^5+34 x^4+82 x^3+46 x^2+7 x+9$
• $y^2=28 x^6+48 x^5+2 x^4+27 x^3+45 x^2+32 x+9$
• $y^2=61 x^6+59 x^5+88 x^4+92 x^3+24 x^2+15 x+95$
• $y^2=55 x^6+56 x^5+36 x^4+59 x^3+24 x^2+80 x+87$
• $y^2=82 x^6+36 x^5+39 x^4+89 x^3+19 x^2+61 x+25$
• $y^2=4 x^6+78 x^5+56 x^4+6 x^3+87 x^2+22 x+61$
• $y^2=84 x^6+73 x^5+85 x^4+21 x^3+72 x^2+31 x+24$
• $y^2=14 x^6+29 x^5+4 x^4+26 x^3+35 x^2+96 x+26$
• $y^2=72 x^6+16 x^5+26 x^4+84 x^3+19 x^2+78 x+86$
• $y^2=5 x^6+65 x^5+87 x^4+10 x^3+74 x^2+91 x+6$
• $y^2=12 x^6+64 x^5+54 x^4+35 x^3+47 x^2+15 x+34$
• $y^2=93 x^6+x^5+35 x^4+95 x^3+87 x^2+59 x+24$
• $y^2=31 x^6+68 x^5+51 x^4+58 x^3+12 x^2+33 x+19$
• $y^2=54 x^6+40 x^5+22 x^4+4 x^3+51 x^2+46 x+87$
• $y^2=63 x^6+12 x^5+78 x^4+85 x^3+18 x^2+28 x+60$
• $y^2=40 x^6+82 x^5+46 x^4+13 x^3+2 x^2+21 x+28$
• $y^2=92 x^6+74 x^5+18 x^4+3 x^3+24 x^2+56 x+48$
• $y^2=15 x^6+40 x^5+65 x^4+8 x^3+8 x^2+51 x+6$
• $y^2=47 x^6+80 x^5+45 x^4+63 x^3+75 x^2+86 x+1$
• $y^2=46 x^6+23 x^5+80 x^4+57 x^3+36 x^2+37 x+18$
• $y^2=9 x^6+49 x^5+44 x^4+8 x^3+39 x^2+95 x+5$
• $y^2=50 x^6+87 x^5+10 x^4+43 x^3+53 x^2+16 x+58$
• $y^2=44 x^6+64 x^5+75 x^4+94 x^3+21 x^2+38 x+74$
• $y^2=82 x^6+17 x^5+41 x^4+8 x^3+66 x^2+81 x+14$
• $y^2=5 x^6+26 x^5+95 x^4+61 x^3+34 x^2+96 x+23$
• $y^2=94 x^6+59 x^5+41 x^4+25 x^3+92 x^2+24 x+14$
• $y^2=51 x^6+47 x^5+90 x^4+42 x^3+72 x^2+49 x+2$
• $y^2=70 x^6+26 x^5+47 x^4+39 x^3+53 x^2+78 x+5$
• $y^2=95 x^6+36 x^5+83 x^4+75 x^3+76 x^2+20 x+60$
• $y^2=55 x^6+68 x^5+17 x^4+24 x^3+83 x^2+27 x+13$
• $y^2=92 x^6+6 x^5+44 x^4+17 x^3+28 x^2+59 x+70$
• $y^2=64 x^6+61 x^5+42 x^4+73 x^3+6 x^2+58 x+93$
• $y^2=70 x^6+91 x^5+32 x^4+31 x^3+6 x^2+55 x+35$
• $y^2=16 x^6+45 x^5+54 x^4+8 x^3+36 x^2+65 x+67$
• $y^2=88 x^6+90 x^5+6 x^4+42 x^3+20 x^2+39 x+21$
• $y^2=21 x^6+37 x^5+10 x^4+47 x^3+31 x^2+50 x+7$
• $y^2=84 x^6+59 x^5+88 x^4+59 x^3+47 x^2+46 x+25$
• $y^2=44 x^6+31 x^5+40 x^4+88 x^3+54 x^2+92 x+21$
• $y^2=32 x^6+13 x^5+68 x^4+39 x^3+35 x^2+50 x+74$
• $y^2=9 x^6+77 x^5+72 x^4+50 x^3+66 x^2+70 x+67$
• $y^2=39 x^6+15 x^5+69 x^4+69 x^3+24 x^2+43 x+94$
• $y^2=39 x^6+9 x^5+54 x^4+58 x^3+86 x^2+73 x+68$
• $y^2=55 x^6+31 x^5+37 x^4+71 x^3+50 x^2+30 x+76$
• $y^2=13 x^6+35 x^5+70 x^4+53 x^3+80 x^2+76 x+62$
• $y^2=27 x^6+10 x^5+x^4+80 x^3+72 x^2+79 x+95$
• $y^2=55 x^6+23 x^5+57 x^4+6 x^3+50 x+39$
• $y^2=61 x^6+87 x^5+74 x^4+2 x^3+71 x^2+31 x+92$
• $y^2=44 x^6+37 x^5+72 x^4+40 x^3+39 x^2+64 x+25$
• $y^2=33 x^6+96 x^5+6 x^4+49 x^3+26 x^2+61 x+32$
• $y^2=27 x^5+88 x^4+70 x^3+85 x^2+66 x+30$
• $y^2=64 x^6+91 x^5+2 x^4+57 x^3+96 x^2+5 x+46$
• $y^2=19 x^6+50 x^5+85 x^4+24 x^3+22 x+82$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ab $\times$ 1.97.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.97.ab : \(\Q(\sqrt{-43}) \).
• 1.97.c : \(\Q(\sqrt{-6}) \).

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.ad_ho	$2$	(not in LMFDB)
2.97.ab_hk	$2$	(not in LMFDB)
2.97.d_ho	$2$	(not in LMFDB)