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

• Label: 2.97.a_hm
• Base field: $\F_{97}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$( 1 + 97 x^{2} )^{2}$
	$1 + 194 x^{2} + 9409 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$63$

This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9604$	$92236816$	$832973830276$	$7834102237495296$	$73742412706667506564$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$9798$	$912674$	$88491646$	$8587340258$	$832975655622$	$80798284478114$	$7837433240259838$	$760231058654565218$	$73742412723842187078$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 63 curves (of which all are hyperelliptic):

• $y^2=82 x^6+11 x^5+86 x^4+42 x^2+81 x+65$
• $y^2=33 x^6+91 x^5+92 x^4+68 x^3+67 x^2+61 x+96$
• $y^2=74 x^6+30 x^5+33 x^4+68 x^2+26 x+35$
• $y^2=44 x^6+81 x^5+78 x^4+44 x^3+31 x^2+69 x+38$
• $y^2=26 x^6+17 x^5+2 x^4+26 x^3+58 x^2+54 x+93$
• $y^2=48 x^6+74 x^5+14 x^4+51 x^3+65 x^2+58 x+17$
• $y^2=67 x^6+26 x^5+14 x^4+70 x^2+29 x+33$
• $y^2=38 x^6+23 x^5+86 x^4+45 x^3+13 x^2+23 x+92$
• $y^2=9 x^6+41 x^5+47 x^4+62 x^3+26 x^2+10 x+14$
• $y^2=82 x^6+90 x^5+32 x^4+36 x^3+96 x+31$
• $y^2=22 x^6+62 x^5+63 x^4+83 x^3+92 x+58$
• $y^2=74 x^6+62 x^5+37 x^3+93 x^2+74 x+39$
• $y^2=79 x^6+19 x^5+88 x^3+77 x^2+79 x+1$
• $y^2=15 x^6+7 x^5+53 x^4+15 x^3+47 x^2+87 x+21$
• $y^2=75 x^6+35 x^5+71 x^4+75 x^3+41 x^2+47 x+8$
• $y^2=29 x^6+87 x^5+78 x^4+70 x^3+78 x^2+87 x+29$
• $y^2=48 x^6+47 x^5+2 x^4+59 x^3+2 x^2+47 x+48$
• $y^2=44 x^6+2 x^5+86 x^4+27 x^3+61 x^2+58 x+86$
• $y^2=26 x^6+10 x^5+42 x^4+38 x^3+14 x^2+96 x+42$
• $y^2=93 x^6+28 x^4+28 x^2+93$
• and

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{2}}$.

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-97}) \)$)$

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

The base change of $A$ to $\F_{97^{2}}$ is 1.9409.hm^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $97$ and $\infty$.

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_adt	$3$	(not in LMFDB)
2.97.a_ahm	$4$	(not in LMFDB)
2.97.a_adt	$6$	(not in LMFDB)