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

• Label: 2.97.bc_pa
• 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:	$( 1 + 14 x + 97 x^{2} )^{2}$
	$1 + 28 x + 390 x^{2} + 2716 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.751640801674$, $\pm0.751640801674$
Angle rank:	$1$ (numerical)
Jacobians:	$62$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12544$	$88510464$	$830547886336$	$7840765305225216$	$73740104688111280384$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$126$	$9406$	$910014$	$88566910$	$8587071486$	$832972117822$	$80798308968510$	$7837433240560894$	$760231061232421758$	$73742412687722993086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 62 curves (of which all are hyperelliptic):

• $y^2=71 x^6+28 x^5+38 x^4+35 x^3+38 x^2+28 x+71$
• $y^2=57 x^6+57 x^5+34 x^4+59 x^3+52 x^2+37 x+68$
• $y^2=40 x^6+94 x^5+13 x^4+8 x^3+74 x^2+79 x+38$
• $y^2=5 x^6+83 x^5+26 x^4+78 x^3+26 x^2+83 x+5$
• $y^2=82 x^6+35 x^5+25 x^4+33 x^3+73 x^2+47 x+21$
• $y^2=55 x^6+45 x^5+74 x^4+41 x^3+74 x^2+45 x+55$
• $y^2=4 x^6+88 x^5+6 x^4+19 x^3+27 x^2+36 x+25$
• $y^2=28 x^6+26 x^5+32 x^4+49 x^3+32 x^2+26 x+28$
• $y^2=74 x^6+56 x^5+73 x^4+28 x^3+22 x^2+49 x+88$
• $y^2=11 x^6+82 x^4+82 x^2+11$
• $y^2=4 x^6+52 x^5+72 x^4+54 x^3+72 x^2+52 x+4$
• $y^2=10 x^6+20 x^5+82 x^4+42 x^3+82 x^2+20 x+10$
• $y^2=19 x^6+42 x^4+42 x^2+19$
• $y^2=47 x^6+26 x^5+21 x^4+36 x^3+21 x^2+26 x+47$
• $y^2=25 x^6+20 x^5+83 x^4+7 x^3+83 x^2+20 x+25$
• $y^2=4 x^6+5 x^5+68 x^4+18 x^3+68 x^2+5 x+4$
• $y^2=3 x^6+90 x^4+90 x^2+3$
• $y^2=66 x^6+91 x^5+21 x^4+50 x^3+51 x^2+24 x+24$
• $y^2=37 x^6+52 x^5+40 x^4+26 x^3+40 x^2+52 x+37$
• $y^2=65 x^6+51 x^5+35 x^4+88 x^3+35 x^2+51 x+65$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.o^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.abc_pa	$2$	(not in LMFDB)
2.97.a_ac	$2$	(not in LMFDB)
2.97.abm_vj	$3$	(not in LMFDB)
2.97.ao_dv	$3$	(not in LMFDB)
2.97.af_acu	$3$	(not in LMFDB)
2.97.k_il	$3$	(not in LMFDB)
2.97.t_ke	$3$	(not in LMFDB)