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

• Label: 2.97.ak_il
• 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:	$( 1 - 5 x + 97 x^{2} )^{2}$
	$1 - 10 x + 219 x^{2} - 970 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.418307468341$, $\pm0.418307468341$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$3, 31$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8649$	$91757241$	$835403312016$	$7835708784772521$	$73739360368158976089$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$9748$	$915334$	$88509796$	$8586984808$	$832972117822$	$80798319521224$	$7837433758641988$	$760231056076708678$	$73742412660669835828$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

• $y^2=64 x^6+94 x^5+61 x^4+89 x^3+3 x^2+41 x+16$
• $y^2=34 x^6+94 x^5+72 x^4+86 x^3+2 x^2+11 x+63$
• $y^2=79 x^6+31 x^5+89 x^4+3 x^3+61 x^2+70 x+33$
• $y^2=57 x^6+x^5+59 x^4+48 x^3+29 x^2+44 x+79$
• $y^2=92 x^6+24 x^5+39 x^4+40 x^3+95 x^2+2 x+27$
• $y^2=34 x^6+62 x^5+48 x^4+28 x^3+62 x^2+93 x+78$
• $y^2=81 x^6+58 x^5+30 x^4+67 x^3+28 x^2+45 x+15$
• $y^2=29 x^6+2 x^5+66 x^4+67 x^3+73 x^2+66 x+40$
• $y^2=69 x^6+94 x^5+66 x^4+22 x^3+15 x^2+73 x+85$
• $y^2=29 x^6+51 x^5+41 x^4+19 x^3+45 x^2+90 x+41$
• $y^2=68 x^6+19 x^5+68 x^4+28 x^3+68 x^2+19 x+68$
• $y^2=x^6+24 x^3+50$
• $y^2=14 x^6+31 x^5+45 x^4+7 x^3+20 x^2+81 x+8$
• $y^2=x^6+65 x^3+18$
• $y^2=69 x^6+73 x^5+70 x^4+66 x^3+50 x^2+61 x+67$
• $y^2=36 x^6+27 x^5+28 x^4+37 x^3+55 x^2+85 x+24$
• $y^2=78 x^6+44 x^5+20 x^4+56 x^3+67 x^2+53 x+40$
• $y^2=67 x^6+81 x^5+42 x^4+61 x^3+11 x^2+64 x+26$
• $y^2=25 x^6+43 x^5+84 x^4+21 x^3+84 x^2+33 x+88$
• $y^2=17 x^6+30 x^5+3 x^4+x^3+25 x^2+14 x+23$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.97.a_gn	$2$	(not in LMFDB)
2.97.k_il	$2$	(not in LMFDB)
2.97.abc_pa	$3$	(not in LMFDB)
2.97.at_ke	$3$	(not in LMFDB)
2.97.f_acu	$3$	(not in LMFDB)
2.97.o_dv	$3$	(not in LMFDB)
2.97.bm_vj	$3$	(not in LMFDB)