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

• Label: 2.97.i_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:	$\pm0.565091650464$, $\pm0.565091650464$
Angle rank:	$1$ (numerical)
Jacobians:	$66$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10404$	$91929744$	$830967157476$	$7835155901485056$	$73745129147852490084$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$106$	$9766$	$910474$	$88503550$	$8587656586$	$832973235622$	$80798248871530$	$7837433617426174$	$760231062016207018$	$73742412673810485286$

Jacobians and polarizations

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

• $y^2=59 x^6+25 x^5+44 x^4+37 x^3+16 x^2+92 x+48$
• $y^2=58 x^6+48 x^5+86 x^4+50 x^3+48 x^2+14 x+73$
• $y^2=18 x^6+70 x^5+53 x^4+35 x^3+53 x^2+70 x+18$
• $y^2=74 x^6+60 x^5+4 x^4+79 x^3+51 x^2+7 x+45$
• $y^2=27 x^6+8 x^5+52 x^4+92 x^3+3 x^2+40 x+58$
• $y^2=89 x^6+13 x^5+13 x^4+60 x^3+23 x^2+78 x+27$
• $y^2=33 x^6+33 x^5+10 x^4+3 x^3+83 x^2+22 x+8$
• $y^2=91 x^6+38 x^4+38 x^2+91$
• $y^2=73 x^6+64 x^5+37 x^4+87 x^3+25 x^2+12 x+26$
• $y^2=18 x^6+58 x^5+78 x^4+64 x^3+15 x^2+31 x+41$
• $y^2=3 x^6+5 x^5+43 x^4+12 x^3+43 x^2+5 x+3$
• $y^2=83 x^6+96 x^5+94 x^4+72 x^3+2 x^2+80 x+58$
• $y^2=47 x^6+44 x^5+88 x^4+7 x^3+4 x^2+84 x+87$
• $y^2=65 x^6+90 x^5+66 x^4+85 x^3+59 x^2+16 x+43$
• $y^2=3 x^6+50 x^5+93 x^4+89 x^3+93 x^2+50 x+3$
• $y^2=82 x^6+54 x^5+x^4+55 x^3+x^2+54 x+82$
• $y^2=49 x^6+50 x^5+39 x^4+53 x^3+21 x^2+x+33$
• $y^2=27 x^6+75 x^5+49 x^4+90 x^3+33 x^2+4 x+64$
• $y^2=86 x^6+84 x^5+19 x^4+46 x^3+36 x^2+67 x+16$
• $y^2=x^6+66 x^3+85$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.e^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.ai_ic	$2$	(not in LMFDB)
2.97.a_gw	$2$	(not in LMFDB)
2.97.ae_add	$3$	(not in LMFDB)