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

• Label: 2.97.abc_ow
• 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:	$( 1 - 16 x + 97 x^{2} )( 1 - 12 x + 97 x^{2} )$
	$1 - 28 x + 386 x^{2} - 2716 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.198227810371$, $\pm0.291487575149$
Angle rank:	$2$ (numerical)
Jacobians:	$96$
Isomorphism classes:	240
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7052$	$88432080$	$835095872492$	$7840204274073600$	$73744215812253662252$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$9398$	$914998$	$88560574$	$8587550230$	$832972230326$	$80798272862566$	$7837433457996286$	$760231057956056806$	$73742412688908952118$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 96 curves (of which all are hyperelliptic):

• $y^2=41 x^6+60 x^5+83 x^4+40 x^3+83 x^2+60 x+41$
• $y^2=63 x^6+x^5+76 x^4+35 x^3+66 x^2+73 x+80$
• $y^2=82 x^6+26 x^5+65 x^4+60 x^3+45 x^2+61 x+33$
• $y^2=19 x^6+71 x^5+38 x^4+10 x^3+90 x^2+33 x+10$
• $y^2=69 x^6+62 x^5+43 x^4+29 x^3+48 x^2+43 x+19$
• $y^2=59 x^6+53 x^5+30 x^4+88 x^3+x^2+92 x+57$
• $y^2=82 x^6+56 x^5+31 x^4+33 x^3+67 x^2+56 x+64$
• $y^2=57 x^6+96 x^5+20 x^4+37 x^3+93 x^2+56 x+50$
• $y^2=33 x^6+34 x^5+51 x^4+18 x^3+8 x^2+29 x+93$
• $y^2=54 x^6+4 x^5+49 x^4+34 x^3+5 x^2+58 x+35$
• $y^2=83 x^6+91 x^5+94 x^4+67 x^3+94 x^2+91 x+83$
• $y^2=3 x^6+95 x^5+64 x^4+80 x^3+64 x^2+95 x+3$
• $y^2=80 x^6+44 x^5+35 x^4+74 x^3+35 x^2+44 x+80$
• $y^2=62 x^6+62 x^5+20 x^4+94 x^3+16 x^2+47 x+70$
• $y^2=35 x^6+7 x^5+3 x^4+88 x^3+91 x^2+28 x+11$
• $y^2=56 x^6+73 x^5+46 x^4+10 x^3+92 x^2+60 x+93$
• $y^2=29 x^6+40 x^5+61 x^4+94 x^3+61 x^2+40 x+29$
• $y^2=35 x^6+68 x^5+3 x^4+10 x^3+62 x^2+19 x+49$
• $y^2=88 x^6+20 x^5+27 x^4+17 x^3+91 x^2+92 x+48$
• $y^2=87 x^6+9 x^5+19 x^4+85 x^3+71 x^2+89 x+83$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.aq $\times$ 1.97.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.97.aq : \(\Q(\sqrt{-33}) \).
• 1.97.am : \(\Q(\sqrt{-61}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.97.ae_c	$2$	(not in LMFDB)
2.97.e_c	$2$	(not in LMFDB)
2.97.bc_ow	$2$	(not in LMFDB)