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

• Label: 2.97.bg_re
• 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:	$( 1 + 14 x + 97 x^{2} )( 1 + 18 x + 97 x^{2} )$
	$1 + 32 x + 446 x^{2} + 3104 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.751640801674$, $\pm0.866875061252$
Angle rank:	$2$ (numerical)
Jacobians:	$62$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12992$	$87306240$	$832301312192$	$7839269196595200$	$73740467446036137152$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$130$	$9278$	$911938$	$88550014$	$8587113730$	$832973533886$	$80798279155906$	$7837433590848766$	$760231058526694786$	$73742412697292441918$

Jacobians and polarizations

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

• $y^2=33 x^6+82 x^5+47 x^4+30 x^3+34 x^2+61 x+72$
• $y^2=83 x^6+59 x^5+49 x^4+48 x^3+33 x^2+51 x+71$
• $y^2=26 x^6+57 x^5+11 x^4+51 x^3+11 x^2+57 x+26$
• $y^2=24 x^6+11 x^5+79 x^4+30 x^3+79 x^2+11 x+24$
• $y^2=43 x^6+54 x^5+88 x^4+52 x^3+18 x^2+86 x+17$
• $y^2=59 x^6+16 x^5+2 x^4+57 x^3+2 x^2+16 x+59$
• $y^2=92 x^6+37 x^5+24 x^4+63 x^3+48 x^2+51 x+57$
• $y^2=43 x^6+18 x^5+87 x^4+72 x^3+11 x^2+72 x+29$
• $y^2=31 x^6+55 x^5+62 x^4+15 x^3+72 x^2+34 x+54$
• $y^2=95 x^6+71 x^5+74 x^4+80 x^3+14 x^2+87 x+66$
• $y^2=79 x^6+87 x^5+32 x^4+54 x^3+70 x^2+59 x+8$
• $y^2=7 x^6+15 x^5+49 x^4+71 x^3+49 x^2+15 x+7$
• $y^2=61 x^6+42 x^5+65 x^4+88 x^3+65 x^2+42 x+61$
• $y^2=62 x^6+55 x^5+61 x^4+80 x^3+61 x^2+55 x+62$
• $y^2=8 x^6+33 x^5+87 x^4+4 x^3+85 x^2+78 x+44$
• $y^2=52 x^6+54 x^5+71 x^4+5 x^3+87 x^2+43 x+77$
• $y^2=45 x^6+96 x^5+5 x^4+76 x^3+90 x^2+64 x+55$
• $y^2=25 x^6+19 x^5+82 x^4+8 x^3+41 x^2+15 x+18$
• $y^2=5 x^6+87 x^5+90 x^4+55 x^3+90 x^2+87 x+5$
• $y^2=48 x^6+59 x^5+20 x^4+56 x^3+17 x^2+71 x+32$
• and

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

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 $\times$ 1.97.s 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.o : \(\Q(\sqrt{-3}) \).
• 1.97.s : \(\Q(\sqrt{-1}) \).

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.abg_re	$2$	(not in LMFDB)
2.97.ae_acg	$2$	(not in LMFDB)
2.97.e_acg	$2$	(not in LMFDB)
2.97.ab_afs	$3$	(not in LMFDB)
2.97.x_ky	$3$	(not in LMFDB)