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

• Label: 2.97.ai_ic
• 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:	$( 1 - 4 x + 97 x^{2} )^{2}$
	$1 - 8 x + 210 x^{2} - 776 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.434908349536$, $\pm0.434908349536$
Angle rank:	$1$ (numerical)
Jacobians:	$66$
Cyclic group of points:	no
Non-cyclic primes:	$2, 47$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8836$	$91929744$	$834982923076$	$7835155901485056$	$73739696315514224836$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$9766$	$914874$	$88503550$	$8587023930$	$832973235622$	$80798320084698$	$7837433617426174$	$760231055292923418$	$73742412673810485286$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 66 curves (of which all are hyperelliptic):

• $y^2=70 x^6+5 x^5+67 x^4+85 x^3+42 x^2+96 x+29$
• $y^2=31 x^6+29 x^5+56 x^4+10 x^3+29 x^2+61 x+34$
• $y^2=23 x^6+14 x^5+30 x^4+7 x^3+30 x^2+14 x+23$
• $y^2=73 x^6+12 x^5+59 x^4+74 x^3+49 x^2+79 x+9$
• $y^2=83 x^6+21 x^5+88 x^4+96 x^3+20 x^2+8 x+31$
• $y^2=57 x^6+65 x^5+65 x^4+9 x^3+18 x^2+2 x+38$
• $y^2=26 x^6+26 x^5+2 x^4+20 x^3+36 x^2+82 x+21$
• $y^2=67 x^6+93 x^4+93 x^2+67$
• $y^2=74 x^6+29 x^5+88 x^4+47 x^3+28 x^2+60 x+33$
• $y^2=23 x^6+31 x^5+35 x^4+71 x^3+3 x^2+45 x+47$
• $y^2=20 x^6+x^5+28 x^4+80 x^3+28 x^2+x+20$
• $y^2=36 x^6+58 x^5+77 x^4+92 x^3+78 x^2+16 x+31$
• $y^2=41 x^6+26 x^5+52 x^4+35 x^3+20 x^2+32 x+47$
• $y^2=34 x^6+62 x^5+39 x^4+37 x^3+4 x^2+80 x+21$
• $y^2=20 x^6+10 x^5+38 x^4+76 x^3+38 x^2+10 x+20$
• $y^2=94 x^6+69 x^5+39 x^4+11 x^3+39 x^2+69 x+94$
• $y^2=51 x^6+56 x^5+x^4+71 x^3+8 x^2+5 x+68$
• $y^2=83 x^6+15 x^5+68 x^4+18 x^3+26 x^2+59 x+71$
• $y^2=42 x^6+32 x^5+95 x^4+36 x^3+83 x^2+44 x+80$
• $y^2=5 x^6+39 x^3+37$
• and

• $y^2=79 x^6+3 x^5+4 x^4+64 x^3+36 x^2+74 x+86$
• $y^2=80 x^6+45 x^5+74 x^4+31 x^3+54 x^2+64 x+32$
• $y^2=88 x^6+49 x^5+x^4+75 x^3+86 x^2+93 x+36$
• $y^2=60 x^6+83 x^5+94 x^4+69 x^3+79 x^2+44 x+19$
• $y^2=24 x^6+40 x^5+x^4+16 x^3+22 x^2+40 x+47$
• $y^2=21 x^6+8 x^5+26 x^4+38 x^3+13 x^2+74 x+8$
• $y^2=54 x^5+33 x^4+36 x^2+33 x$
• $y^2=19 x^6+3 x^5+96 x^4+56 x^3+77 x^2+82 x+76$
• $y^2=43 x^6+43 x^5+60 x^4+96 x^3+42 x^2+80 x+37$
• $y^2=59 x^6+46 x^5+86 x^4+49 x^3+71 x^2+51 x+10$
• $y^2=27 x^6+31 x^5+15 x^4+19 x^3+19 x^2+6 x+69$
• $y^2=62 x^6+57 x^5+37 x^4+34 x^3+63 x^2+74 x+65$
• $y^2=85 x^6+35 x^5+22 x^4+7 x^3+71 x^2+16 x+54$
• $y^2=44 x^6+24 x^5+59 x^4+13 x^3+69 x^2+18 x+19$
• $y^2=40 x^6+25 x^5+54 x^4+17 x^3+44 x^2+11 x+72$
• $y^2=72 x^6+86 x^5+44 x^4+26 x^3+95 x^2+12 x+76$
• $y^2=5 x^6+20 x^5+44 x^4+52 x^3+61 x^2+47 x+76$
• $y^2=26 x^6+9 x^5+62 x^4+57 x^3+7 x^2+26 x+93$
• $y^2=10 x^6+9 x^5+86 x^4+5 x^3+19 x^2+58 x+15$
• $y^2=63 x^6+9 x^5+92 x^4+51 x^3+27 x^2+72 x+83$
• $y^2=94 x^6+61 x^5+46 x^4+46 x^3+95 x^2+37 x+32$
• $y^2=41 x^6+32 x^5+2 x^4+74 x^3+54 x^2+7 x+57$
• $y^2=81 x^6+41 x^5+84 x^4+45 x^3+84 x^2+41 x+81$
• $y^2=14 x^6+96 x^5+36 x^4+22 x^3+66 x^2+53 x+43$
• $y^2=5 x^6+38 x^5+80 x^4+25 x^3+11 x^2+79 x+21$
• $y^2=58 x^6+9 x^5+45 x^4+72 x^3+46 x^2+72 x+87$
• $y^2=58 x^6+35 x^5+62 x^4+11 x^3+49 x^2+38 x+24$
• $y^2=43 x^6+83 x^5+95 x^4+26 x^3+81 x^2+54 x+21$
• $y^2=58 x^6+20 x^5+57 x^4+51 x^3+80 x^2+52 x+14$
• $y^2=33 x^6+93 x^5+33 x^4+54 x^3+44 x^2+23 x+69$
• $y^2=94 x^6+79 x^5+49 x^4+73 x^3+52 x^2+4 x+40$
• $y^2=50 x^6+53 x^5+54 x^4+23 x^3+67 x^2+45 x+84$
• $y^2=32 x^6+65 x^5+29 x^4+34 x^3+39 x^2+91 x+13$
• $y^2=2 x^6+64 x^5+74 x^4+26 x^3+63 x^2+77 x+41$
• $y^2=87 x^6+79 x^5+49 x^4+40 x^3+26 x^2+18 x+65$
• $y^2=54 x^6+75 x^5+57 x^4+45 x^3+79 x^2+47 x+75$
• $y^2=84 x^6+66 x^5+14 x^4+61 x^3+20 x^2+3 x+21$
• $y^2=64 x^6+61 x^5+30 x^4+16 x^3+7 x^2+52 x+16$
• $y^2=51 x^6+12 x^5+81 x^4+66 x^3+63 x^2+44 x+84$
• $y^2=21 x^6+51 x^5+47 x^4+17 x^3+85 x^2+43 x+93$
• $y^2=5 x^6+20 x^3+29$
• $y^2=18 x^6+15 x^4+15 x^2+18$
• $y^2=45 x^6+90 x^5+54 x^4+80 x^3+21 x^2+60 x+58$
• $y^2=35 x^6+75 x^5+89 x^4+50 x^3+62 x^2+39 x+7$
• $y^2=95 x^6+18 x^5+90 x^4+34 x^3+90 x^2+18 x+95$
• $y^2=9 x^6+6 x^5+71 x^4+34 x^3+41 x^2+x+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-93}) \)$)$

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_gw	$2$	(not in LMFDB)
2.97.i_ic	$2$	(not in LMFDB)
2.97.e_add	$3$	(not in LMFDB)