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

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

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - 93 x^{2} - 194 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.134291403488$, $\pm0.800958070154$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$77$
Cyclic group of points:	yes

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9121$	$86768073$	$831926410000$	$7838963767566729$	$73741637750289541921$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$96$	$9220$	$911526$	$88546564$	$8587250016$	$832974996670$	$80798296223328$	$7837433715985924$	$760231061419572582$	$73742412680461764100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 77 curves (of which all are hyperelliptic):

• $y^2=54 x^6+47 x^5+65 x^4+52 x^3+64 x^2+94 x+52$
• $y^2=5 x^6+69 x^5+5 x^4+52 x^3+80 x^2+74 x+35$
• $y^2=22 x^6+15 x^5+41 x^4+90 x^3+15 x^2+71 x+32$
• $y^2=21 x^6+49 x^5+57 x^4+66 x^3+86 x^2+59 x+66$
• $y^2=47 x^6+26 x^5+85 x^4+36 x^3+75 x^2+4 x+12$
• $y^2=21 x^6+3 x^5+43 x^4+81 x^3+50 x^2+6 x+78$
• $y^2=17 x^6+18 x^5+38 x^4+39 x^3+53 x^2+30 x+22$
• $y^2=83 x^6+96 x^5+83 x^4+21 x^3+10 x^2+46 x+5$
• $y^2=69 x^6+13 x^5+9 x^4+74 x^3+25 x^2+80 x+67$
• $y^2=86 x^6+41 x^5+69 x^4+74 x^3+44 x^2+95 x+83$
• $y^2=8 x^6+91 x^5+6 x^4+17 x^3+10 x^2+39 x+66$
• $y^2=23 x^6+48 x^5+75 x^4+70 x^3+12 x^2+35 x+34$
• $y^2=27 x^6+89 x^5+82 x^4+86 x^3+86 x^2+24 x+70$
• $y^2=x^6+27 x^5+42 x^4+14 x^3+81 x^2+7 x+38$
• $y^2=43 x^6+82 x^5+53 x^4+78 x^3+39 x^2+96 x+88$
• $y^2=82 x^6+66 x^5+23 x^4+16 x^3+64 x^2+2 x+78$
• $y^2=41 x^6+5 x^5+36 x^4+65 x^3+52 x^2+81 x+38$
• $y^2=91 x^6+85 x^5+81 x^4+22 x^3+43 x^2+71 x+88$
• $y^2=94 x^6+40 x^5+33 x^4+9 x^3+62 x^2+4 x+57$
• $y^2=59 x^6+33 x^5+51 x^4+63 x^3+46 x^2+66 x+10$
• and

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{3}}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\).

Endomorphism algebra over $\overline{\F}_{97}$

The base change of $A$ to $\F_{97^{3}}$ is 1.912673.awc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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.c_adp	$2$	(not in LMFDB)
2.97.e_hq	$3$	(not in LMFDB)
2.97.ae_hq	$6$	(not in LMFDB)