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

• Label: 2.97.ag_hv
• 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:	$( 1 - 3 x + 97 x^{2} )^{2}$
	$1 - 6 x + 203 x^{2} - 582 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.451331388283$, $\pm0.451331388283$
Angle rank:	$1$ (numerical)
Jacobians:	$51$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9025$	$92064025$	$834518790400$	$7834706067515625$	$73740209503583400625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$92$	$9780$	$914366$	$88498468$	$8587083692$	$832974224190$	$80798316022796$	$7837433473742788$	$760231055232828542$	$73742412690929130900$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 51 curves (of which all are hyperelliptic):

• $y^2=67 x^6+20 x^5+12 x^4+3 x^3+6 x^2+37 x+36$
• $y^2=24 x^6+6 x^5+19 x^4+41 x^3+58 x^2+70 x+17$
• $y^2=20 x^6+71 x^5+12 x^4+60 x^3+73 x^2+72 x+75$
• $y^2=82 x^6+56 x^5+80 x^4+59 x^3+13 x^2+76 x+55$
• $y^2=67 x^6+63 x^5+65 x^4+95 x^3+94 x^2+54 x+62$
• $y^2=29 x^6+96 x^5+28 x^4+41 x^3+24 x^2+22 x+28$
• $y^2=23 x^6+36 x^4+15 x^3+62 x^2+23$
• $y^2=28 x^6+18 x^5+13 x^4+67 x^3+39 x^2+65 x+77$
• $y^2=10 x^6+72 x^5+90 x^4+91 x^3+28 x^2+17 x+61$
• $y^2=35 x^6+30 x^5+88 x^4+29 x^3+80 x^2+89 x+52$
• $y^2=52 x^6+28 x^5+80 x^4+76 x^3+82 x^2+85 x+87$
• $y^2=70 x^6+21 x^5+63 x^4+57 x^3+49 x^2+90 x+95$
• $y^2=75 x^6+87 x^5+26 x^4+39 x^3+30 x^2+59 x+17$
• $y^2=50 x^6+52 x^5+54 x^4+81 x^3+6 x^2+61 x+37$
• $y^2=86 x^6+51 x^5+18 x^4+61 x^3+24 x^2+26 x+35$
• $y^2=63 x^6+82 x^5+92 x^4+37 x^3+15 x^2+59 x+45$
• $y^2=21 x^6+69 x^5+87 x^4+x^3+12 x^2+61 x+65$
• $y^2=34 x^6+41 x^5+94 x^4+29 x^3+41 x^2+79 x+20$
• $y^2=21 x^6+67 x^5+44 x^4+82 x^3+81 x^2+20 x+76$
• $y^2=38 x^6+47 x^5+6 x^4+59 x^3+93 x^2+64 x+57$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ad^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-379}) \)$)$

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.a_hd	$2$	(not in LMFDB)
2.97.g_hv	$2$	(not in LMFDB)
2.97.d_adk	$3$	(not in LMFDB)