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

• Label: 2.97.c_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:	$\pm0.199041929846$, $\pm0.865708596512$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$77$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9513$	$86768073$	$834021909504$	$7838963767566729$	$73743187627808666793$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$100$	$9220$	$913822$	$88546564$	$8587430500$	$832974996670$	$80798272732900$	$7837433715985924$	$760231055889557854$	$73742412680461764100$

Jacobians and polarizations

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

• $y^2=69 x^6+87 x^5+13 x^4+88 x^3+71 x^2+77 x+88$
• $y^2=25 x^6+54 x^5+25 x^4+66 x^3+12 x^2+79 x+78$
• $y^2=82 x^6+3 x^5+47 x^4+18 x^3+3 x^2+53 x+84$
• $y^2=43 x^6+68 x^5+89 x^4+52 x^3+56 x^2+70 x+52$
• $y^2=41 x^6+33 x^5+37 x^4+83 x^3+84 x^2+20 x+60$
• $y^2=8 x^6+15 x^5+21 x^4+17 x^3+56 x^2+30 x+2$
• $y^2=81 x^6+23 x^5+27 x^4+66 x^3+30 x^2+6 x+82$
• $y^2=36 x^6+58 x^5+36 x^4+43 x^3+2 x^2+48 x+1$
• $y^2=54 x^6+65 x^5+45 x^4+79 x^3+28 x^2+12 x+44$
• $y^2=56 x^6+47 x^5+72 x^4+73 x^3+67 x^2+19 x+36$
• $y^2=21 x^6+57 x^5+40 x^4+81 x^3+2 x^2+66 x+52$
• $y^2=18 x^6+46 x^5+84 x^4+59 x^3+60 x^2+78 x+73$
• $y^2=83 x^6+76 x^5+94 x^4+56 x^3+56 x^2+63 x+14$
• $y^2=5 x^6+38 x^5+16 x^4+70 x^3+17 x^2+35 x+93$
• $y^2=21 x^6+22 x^5+71 x^4+2 x^3+x^2+92 x+52$
• $y^2=22 x^6+39 x^5+18 x^4+80 x^3+29 x^2+10 x+2$
• $y^2=11 x^6+25 x^5+83 x^4+34 x^3+66 x^2+17 x+93$
• $y^2=67 x^6+37 x^5+17 x^4+13 x^3+21 x^2+64 x+52$
• $y^2=82 x^6+6 x^5+68 x^4+45 x^3+19 x^2+20 x+91$
• $y^2=4 x^6+68 x^5+61 x^4+24 x^3+36 x^2+39 x+50$
• and

• $y^2=75 x^6+60 x^5+34 x^4+70 x^3+38 x^2+28 x+53$
• $y^2=67 x^6+72 x^5+74 x^4+89 x^3+93 x^2+49 x+18$
• $y^2=61 x^6+41 x^5+66 x^4+x^3+82 x^2+47 x+53$
• $y^2=10 x^6+41 x^5+80 x^4+37 x^3+73 x^2+56 x+40$
• $y^2=6 x^6+54 x^5+10 x^4+86 x^3+33 x^2+59 x+6$
• $y^2=51 x^6+2 x^5+75 x^4+57 x^3+53 x^2+77 x+19$
• $y^2=58 x^6+77 x^5+65 x^4+59 x^3+85 x^2+69 x+42$
• $y^2=15 x^6+90 x^5+30 x^4+90 x^3+60 x^2+12 x+63$
• $y^2=35 x^6+26 x^5+12 x^4+92 x^3+67 x^2+65 x+51$
• $y^2=84 x^6+75 x^5+18 x^4+29 x^3+4 x^2+30 x+90$
• $y^2=65 x^6+94 x^5+14 x^4+7 x^3+78 x^2+7 x+61$
• $y^2=79 x^6+25 x^5+34 x^4+42 x^3+73 x^2+4 x+28$
• $y^2=21 x^6+42 x^5+91 x^4+77 x^3+47 x^2+64 x+82$
• $y^2=17 x^6+39 x^5+68 x^4+92 x^3+90 x^2+18 x+75$
• $y^2=93 x^6+54 x^5+56 x^4+28 x^3+2 x^2+77 x+63$
• $y^2=80 x^6+25 x^5+12 x^4+59 x^3+67 x^2+6 x+2$
• $y^2=78 x^6+36 x^5+11 x^4+6 x^3+53 x^2+8 x+19$
• $y^2=57 x^6+9 x^5+84 x^4+25 x^3+94 x^2+25 x+38$
• $y^2=76 x^6+37 x^5+6 x^4+80 x^3+45 x^2+47 x+58$
• $y^2=29 x^6+4 x^5+59 x^4+12 x^3+34 x^2+71 x+50$
• $y^2=52 x^6+72 x^5+43 x^4+79 x^3+34 x^2+72 x+75$
• $y^2=66 x^6+94 x^5+17 x^4+95 x^3+78 x^2+41 x+8$
• $y^2=9 x^6+77 x^5+45 x^4+80 x^3+28 x^2+6 x+89$
• $y^2=54 x^6+x^5+38 x^4+11 x^3+26 x^2+64 x+68$
• $y^2=x^6+x^3+49$
• $y^2=87 x^6+74 x^5+27 x^4+51 x^3+76 x^2+3 x+84$
• $y^2=x^6+5 x^3+6$
• $y^2=75 x^6+36 x^5+8 x^4+69 x^3+83 x+37$
• $y^2=49 x^6+84 x^5+6 x^4+91 x^3+10 x+94$
• $y^2=33 x^6+20 x^5+63 x^4+10 x^3+32 x^2+79 x+13$
• $y^2=32 x^6+60 x^5+91 x^4+62 x^3+30 x^2+29 x+91$
• $y^2=73 x^6+10 x^5+31 x^4+40 x^3+75 x^2+8 x+88$
• $y^2=48 x^6+59 x^5+50 x^4+52 x^3+56 x^2+90 x+29$
• $y^2=90 x^6+38 x^5+62 x^4+53 x^3+49 x^2+75 x+82$
• $y^2=54 x^6+68 x^5+25 x^4+62 x^3+9 x^2+35 x+95$
• $y^2=47 x^6+82 x^5+51 x^4+36 x^3+59 x^2+62 x+62$
• $y^2=33 x^6+79 x^5+54 x^4+80 x^3+14 x^2+4 x+93$
• $y^2=50 x^6+9 x^5+23 x^4+25 x^3+35 x^2+93 x+57$
• $y^2=74 x^6+52 x^5+65 x^4+69 x^3+29 x^2+8 x+92$
• $y^2=73 x^6+88 x^5+54 x^4+76 x^3+75 x+48$
• $y^2=69 x^6+30 x^5+57 x^4+71 x^3+45 x^2+70 x+36$
• $y^2=x^6+5 x^3+86$
• $y^2=43 x^6+32 x^5+30 x^4+55 x^3+33 x^2+21 x+25$
• $y^2=33 x^6+16 x^5+73 x^4+93 x^3+22 x^2+25 x+64$
• $y^2=x^6+x^3+35$
• $y^2=24 x^6+42 x^5+87 x^4+6 x^3+68 x^2+3 x+82$
• $y^2=x^6+x^3+32$
• $y^2=28 x^6+67 x^5+30 x^4+95 x^3+19 x^2+26 x+72$
• $y^2=x^6+x^3+43$
• $y^2=34 x^6+5 x^5+64 x^4+7 x^3+96 x^2+24 x+89$
• $y^2=32 x^6+94 x^5+45 x^4+46 x^3+76 x^2+12 x+93$
• $y^2=85 x^6+51 x^5+60 x^4+46 x^3+35 x^2+41 x+31$
• $y^2=37 x^6+4 x^5+96 x^4+4 x^3+9 x^2+91 x+36$
• $y^2=74 x^6+39 x^5+38 x^4+55 x^3+72 x^2+51 x+67$
• $y^2=x^6+x^3+81$
• $y^2=25 x^6+19 x^5+69 x^4+63 x^3+60 x^2+83 x+58$
• $y^2=3 x^6+20 x^5+49 x^4+21 x^3+10 x^2+5 x+73$

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.wc^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.ac_adp	$2$	(not in LMFDB)
2.97.ae_hq	$3$	(not in LMFDB)
2.97.ac_adp	$6$	(not in LMFDB)