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

• Label: 2.97.h_abw
• 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:	$1 + 7 x - 48 x^{2} + 679 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.282312186075$, $\pm0.948978852742$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{113})\)
Galois group:	$C_2^2$
Jacobians:	$138$
Isomorphism classes:	234

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10048$	$87176448$	$836068839424$	$7837628894843904$	$73743956433617819968$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$105$	$9265$	$916062$	$88531489$	$8587520025$	$832969916350$	$80798274350745$	$7837433422189249$	$760231059100483614$	$73742412704634319825$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 138 curves (of which all are hyperelliptic):

• $y^2=74 x^6+37 x^5+79 x^4+84 x^3+81 x^2+62 x+10$
• $y^2=2 x^6+17 x^5+54 x^4+78 x^3+62 x+20$
• $y^2=70 x^6+34 x^5+5 x^4+33 x^3+9 x^2+71 x+76$
• $y^2=40 x^6+85 x^5+37 x^4+39 x^3+52 x^2+79 x+87$
• $y^2=70 x^6+36 x^5+93 x^4+28 x^3+92 x^2+47 x+30$
• $y^2=3 x^6+x^5+94 x^4+84 x^3+17 x^2+47 x+17$
• $y^2=76 x^6+21 x^5+38 x^4+73 x^3+6 x^2+13 x+64$
• $y^2=21 x^6+42 x^5+14 x^4+85 x^3+24 x^2+35 x+10$
• $y^2=24 x^6+71 x^5+10 x^4+69 x^3+70 x^2+15 x+11$
• $y^2=54 x^6+35 x^5+32 x^4+20 x^3+90 x^2+87 x$
• $y^2=30 x^6+19 x^5+46 x^4+74 x^3+38 x^2+68 x+71$
• $y^2=48 x^6+89 x^5+34 x^4+16 x^3+10 x^2+84 x+54$
• $y^2=11 x^6+79 x^5+59 x^4+69 x^3+50 x^2+31 x+59$
• $y^2=21 x^6+45 x^5+2 x^4+17 x^3+47 x^2+5 x+43$
• $y^2=71 x^6+82 x^5+19 x^4+69 x^3+7 x^2+x+36$
• $y^2=90 x^6+57 x^5+55 x^4+22 x^3+14 x^2+79 x+66$
• $y^2=5 x^6+5 x^3+15$
• $y^2=16 x^6+34 x^5+20 x^4+80 x^3+61 x^2+78 x+12$
• $y^2=30 x^6+15 x^5+86 x^4+70 x^3+23 x^2+87 x+54$
• $y^2=79 x^6+53 x^5+33 x^4+73 x^3+69 x^2+x+53$
• and

• $y^2=79 x^6+54 x^5+72 x^4+61 x^3+71 x^2+80 x+75$
• $y^2=91 x^6+45 x^5+59 x^4+52 x^3+32 x^2+73 x+4$
• $y^2=33 x^6+89 x^5+12 x^4+89 x^3+16 x^2+60 x+48$
• $y^2=24 x^6+29 x^5+6 x^4+6 x^3+82 x^2+86 x+35$
• $y^2=65 x^6+12 x^5+10 x^4+8 x^3+10 x^2+21 x+69$
• $y^2=86 x^6+90 x^5+82 x^4+69 x^3+x^2+29 x+21$
• $y^2=75 x^6+73 x^5+52 x^4+33 x^3+4 x^2+79 x+29$
• $y^2=7 x^6+21 x^5+47 x^4+69 x^3+18 x^2+60 x+11$
• $y^2=52 x^6+20 x^5+11 x^4+44 x^3+22 x^2+58 x+18$
• $y^2=16 x^6+16 x^5+68 x^4+11 x^3+32 x^2+87 x+12$
• $y^2=43 x^6+51 x^5+15 x^4+87 x^3+21 x^2+89 x+9$
• $y^2=12 x^6+13 x^5+2 x^4+48 x^3+96 x^2+25 x+41$
• $y^2=94 x^6+63 x^5+85 x^4+95 x^3+89 x^2+34 x+1$
• $y^2=19 x^6+72 x^5+17 x^4+9 x^3+59 x^2+23 x+12$
• $y^2=89 x^6+42 x^5+45 x^4+29 x^3+41 x^2+63 x+84$
• $y^2=22 x^6+71 x^5+63 x^4+90 x^3+31 x^2+33 x+87$
• $y^2=67 x^6+92 x^5+67 x^4+21 x^3+5 x^2+19 x+59$
• $y^2=67 x^6+10 x^5+47 x^4+36 x^3+3 x^2+47 x+82$
• $y^2=5 x^6+44 x^5+41 x^3+45 x^2+93 x+63$
• $y^2=67 x^6+51 x^5+12 x^4+55 x^3+17 x^2+21 x+28$
• $y^2=50 x^6+8 x^5+15 x^4+85 x^3+36 x^2+49 x+60$
• $y^2=5 x^6+5 x^3+14$
• $y^2=72 x^6+69 x^5+6 x^4+52 x^3+51 x^2+46 x+88$
• $y^2=5 x^6+5 x^3+87$
• $y^2=11 x^6+9 x^5+43 x^4+92 x^3+96 x^2+78 x+84$
• $y^2=75 x^6+36 x^5+29 x^4+81 x^3+71 x^2+29 x+29$
• $y^2=47 x^6+40 x^5+12 x^4+89 x^3+19 x^2+12 x+62$
• $y^2=80 x^6+58 x^5+89 x^4+43 x^3+51 x^2+21 x+46$
• $y^2=76 x^6+92 x^5+36 x^4+86 x^3+7 x^2+88 x+70$
• $y^2=56 x^6+28 x^5+46 x^4+37 x^3+26 x^2+96 x+55$
• $y^2=33 x^6+68 x^5+70 x^4+78 x^3+20 x^2+80 x+41$
• $y^2=83 x^6+16 x^5+28 x^4+58 x^3+5 x^2+72 x+30$
• $y^2=50 x^6+20 x^5+20 x^4+41 x^3+84 x^2+9 x+36$
• $y^2=79 x^6+32 x^5+53 x^4+22 x^3+52 x^2+41 x+47$
• $y^2=57 x^6+26 x^5+42 x^4+20 x^3+29 x^2+29 x+44$
• $y^2=39 x^6+63 x^5+28 x^4+73 x^2+60 x+9$
• $y^2=93 x^6+70 x^5+80 x^4+52 x^3+7 x^2+28 x$
• $y^2=12 x^6+69 x^5+34 x^4+29 x^3+9 x^2+50 x+86$
• $y^2=85 x^6+65 x^5+76 x^4+58 x^3+4 x^2+4 x+70$
• $y^2=30 x^6+14 x^5+79 x^4+12 x^3+3 x^2+90 x+44$
• $y^2=51 x^6+38 x^5+28 x^4+9 x^3+23 x^2+5 x+5$
• $y^2=11 x^6+7 x^5+25 x^3+87 x^2+40 x+95$
• $y^2=23 x^6+32 x^5+29 x^4+80 x^3+49 x^2+89 x+19$
• $y^2=47 x^6+88 x^5+62 x^3+48 x^2+84 x+28$
• $y^2=27 x^6+26 x^5+75 x^4+5 x^3+8 x^2+82 x+51$
• $y^2=4 x^6+36 x^5+75 x^4+52 x^3+14 x^2+53 x+7$
• $y^2=44 x^6+71 x^5+61 x^4+15 x^3+9 x^2+83 x+59$
• $y^2=22 x^6+14 x^5+63 x^4+86 x^3+38 x^2+37 x+66$
• $y^2=18 x^6+37 x^5+43 x^4+86 x^3+72 x^2+21 x+90$
• $y^2=49 x^6+26 x^5+57 x^4+80 x^3+48 x^2+15 x+83$
• $y^2=76 x^6+x^5+60 x^4+29 x^3+55 x^2+36 x+69$
• $y^2=82 x^6+71 x^5+11 x^4+29 x^3+66 x^2+42 x+94$
• $y^2=27 x^5+76 x^4+78 x^3+73 x^2+57 x+79$
• $y^2=60 x^6+74 x^5+96 x^4+50 x^3+3 x^2+29 x+12$
• $y^2=94 x^6+31 x^5+27 x^4+53 x^3+61 x^2+55 x+67$
• $y^2=26 x^6+75 x^5+63 x^4+10 x^3+66 x^2+43 x+25$
• $y^2=20 x^6+59 x^5+56 x^4+95 x^3+31 x^2+19 x+1$
• $y^2=8 x^6+41 x^5+5 x^4+61 x^3+72 x^2+15 x+63$
• $y^2=40 x^6+3 x^5+78 x^4+49 x^3+42 x^2+57 x+21$
• $y^2=11 x^6+8 x^5+62 x^4+43 x^3+90 x^2+96 x+15$
• $y^2=74 x^5+49 x^4+21 x^3+80 x^2+66 x+35$
• $y^2=82 x^6+91 x^5+67 x^4+71 x^3+12 x^2+30 x+82$
• $y^2=75 x^6+7 x^5+46 x^4+51 x^3+75 x^2+96 x+19$
• $y^2=46 x^6+56 x^5+12 x^4+30 x^3+14 x^2+7 x+50$
• $y^2=75 x^6+35 x^5+43 x^3+57 x^2+87 x+61$
• $y^2=4 x^6+54 x^5+82 x^4+81 x^3+29 x^2+38 x+80$
• $y^2=72 x^6+51 x^5+3 x^4+65 x^3+27 x^2+72 x+95$
• $y^2=4 x^6+74 x^5+50 x^4+39 x^3+12 x^2+54 x+77$
• $y^2=84 x^6+2 x^5+6 x^4+50 x^3+31 x^2+13 x+43$
• $y^2=91 x^6+43 x^5+46 x^4+48 x^3+20 x^2+77 x+55$
• $y^2=33 x^6+53 x^4+52 x^3+17 x^2+25 x+18$
• $y^2=54 x^6+37 x^5+5 x^4+29 x^3+76 x^2+12 x+19$
• $y^2=4 x^6+65 x^5+78 x^4+86 x^3+63 x^2+80 x+75$
• $y^2=69 x^6+68 x^5+6 x^4+40 x^3+24 x^2+63 x+65$
• $y^2=95 x^6+37 x^5+46 x^4+64 x^3+75 x^2+43 x+60$
• $y^2=66 x^6+36 x^5+82 x^4+45 x^3+34 x^2+17 x+68$
• $y^2=33 x^6+58 x^5+2 x^4+42 x^3+27 x^2+34 x+79$
• $y^2=37 x^6+24 x^5+4 x^4+25 x^3+24 x^2+36 x+11$
• $y^2=64 x^6+69 x^5+72 x^4+78 x^3+59 x^2+37 x+77$
• $y^2=79 x^6+69 x^5+23 x^4+91 x^3+10 x^2+66 x+81$
• $y^2=25 x^6+14 x^5+79 x^4+60 x^3+2 x^2+53 x+31$
• $y^2=9 x^6+36 x^5+77 x^4+57 x^3+70 x^2+88 x+90$
• $y^2=84 x^6+70 x^5+55 x^4+24 x^3+82 x^2+78 x+75$
• $y^2=71 x^6+27 x^5+x^4+52 x^3+55 x^2+89$
• $y^2=7 x^6+4 x^5+92 x^4+79 x^3+56 x^2+40 x+47$
• $y^2=95 x^6+48 x^5+38 x^4+71 x^3+24 x^2+31 x+60$
• $y^2=46 x^6+46 x^5+64 x^4+73 x^3+65 x^2+44 x+53$
• $y^2=30 x^6+17 x^5+61 x^4+96 x^3+21 x^2+12 x+83$
• $y^2=42 x^6+53 x^5+17 x^4+31 x^3+46 x^2+46 x+6$
• $y^2=58 x^6+47 x^5+75 x^4+93 x^3+91 x^2+90 x+76$
• $y^2=59 x^6+73 x^5+35 x^4+58 x^3+36 x^2+75 x+19$
• $y^2=62 x^6+22 x^5+65 x^4+21 x^3+46 x^2+27 x+86$
• $y^2=46 x^6+45 x^5+12 x^4+27 x^3+91 x^2+64 x+85$
• $y^2=77 x^6+2 x^5+23 x^4+50 x^3+13 x^2+48 x+74$
• $y^2=53 x^6+66 x^5+32 x^4+38 x^3+77 x^2+44 x+36$
• $y^2=94 x^6+45 x^5+75 x^4+93 x^3+88 x^2+20 x+91$
• $y^2=34 x^6+39 x^5+13 x^4+17 x^3+63 x^2+90 x+20$
• $y^2=x^6+4 x^5+25 x^4+29 x^3+30 x^2+64 x+38$
• $y^2=72 x^6+77 x^5+41 x^4+18 x^3+39 x^2+54 x+1$
• $y^2=64 x^6+62 x^5+63 x^4+96 x^3+x^2+79 x+57$
• $y^2=10 x^6+53 x^5+x^4+7 x^3+24 x^2+9 x+45$
• $y^2=46 x^6+90 x^5+55 x^4+84 x^3+3 x^2+63 x+26$
• $y^2=62 x^5+26 x^4+27 x^3+52 x^2+70 x+74$
• $y^2=9 x^6+85 x^5+79 x^4+61 x^3+2 x^2+27 x+28$
• $y^2=69 x^6+83 x^5+88 x^4+33 x^3+77 x^2+47 x+36$
• $y^2=46 x^6+38 x^5+76 x^4+32 x^3+57 x+39$
• $y^2=90 x^6+91 x^5+37 x^4+24 x^3+18 x^2+72 x+20$
• $y^2=54 x^6+33 x^5+38 x^4+11 x^3+77 x^2+76 x+10$
• $y^2=37 x^6+86 x^5+26 x^4+41 x^3+39 x^2+18 x+96$
• $y^2=10 x^6+82 x^5+83 x^4+48 x^3+19 x^2+32 x+2$
• $y^2=65 x^6+8 x^5+37 x^4+84 x^3+34 x^2+33 x+92$
• $y^2=76 x^6+11 x^5+64 x^4+3 x^3+75 x^2+24 x+83$
• $y^2=92 x^6+96 x^5+89 x^4+96 x^3+27 x^2+35 x+4$
• $y^2=24 x^6+77 x^5+54 x^4+82 x^3+24 x^2+16 x+95$
• $y^2=23 x^6+73 x^5+13 x^4+75 x^3+94 x^2+22 x$
• $y^2=57 x^6+70 x^5+49 x^4+80 x^3+26 x^2+5 x+3$
• $y^2=66 x^6+73 x^5+73 x^4+20 x^3+8 x^2+5 x+89$
• $y^2=21 x^6+16 x^5+6 x^4+64 x^3+13 x^2+22 x+32$

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 \(\Q(\sqrt{-3}, \sqrt{113})\).

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

The base change of $A$ to $\F_{97^{3}}$ is 1.912673.cne^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-339}) \)$)$

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.ah_abw	$2$	(not in LMFDB)
2.97.ao_jj	$3$	(not in LMFDB)
2.97.a_fp	$6$	(not in LMFDB)