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

• Label: 2.97.bh_rs
• 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:	$( 1 + 14 x + 97 x^{2} )( 1 + 19 x + 97 x^{2} )$
	$1 + 33 x + 460 x^{2} + 3201 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.751640801674$, $\pm0.915025864992$
Angle rank:	$1$ (numerical)
Jacobians:	$30$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13104$	$86958144$	$832972061376$	$7838296141568256$	$73741630849274881584$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$131$	$9241$	$912674$	$88539025$	$8587249211$	$832972117822$	$80798291446955$	$7837433512244449$	$760231058654565218$	$73742412703904321161$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

• $y^2=38 x^6+94 x^5+91 x^4+72 x^3+33 x^2+54 x+4$
• $y^2=45 x^6+59 x^5+21 x^4+83 x^3+90 x^2+83 x+66$
• $y^2=77 x^6+70 x^5+6 x^4+57 x^3+25 x^2+81 x+65$
• $y^2=50 x^6+54 x^5+41 x^4+23 x^3+60 x^2+52 x+3$
• $y^2=15 x^6+32 x^5+70 x^4+26 x^3+85 x^2+30 x+46$
• $y^2=27 x^6+51 x^5+15 x^4+85 x^3+38 x^2+93 x+79$
• $y^2=23 x^6+96 x^5+51 x^4+82 x^3+64 x^2+33 x+93$
• $y^2=12 x^6+56 x^5+47 x^4+2 x^3+43 x^2+58 x+82$
• $y^2=18 x^6+6 x^5+50 x^4+23 x^3+15 x^2+43 x+44$
• $y^2=22 x^6+6 x^5+91 x^4+91 x^3+22 x^2+50 x+91$
• $y^2=75 x^6+26 x^5+80 x^4+7 x^3+30 x^2+40 x+19$
• $y^2=72 x^6+27 x^5+66 x^4+90 x^3+73 x^2+7 x+9$
• $y^2=90 x^6+76 x^5+12 x^4+18 x^3+7 x^2+40 x+47$
• $y^2=64 x^6+31 x^5+96 x^4+57 x^3+5 x^2+53 x+70$
• $y^2=29 x^6+85 x^5+61 x^4+85 x^3+30 x^2+26 x+60$
• $y^2=5 x^6+5 x^3+18$
• $y^2=72 x^6+64 x^5+83 x^4+35 x^3+27 x^2+86 x+24$
• $y^2=50 x^6+85 x^5+67 x^4+51 x^3+29 x^2+53 x+18$
• $y^2=x^6+67 x^5+9 x^4+51 x^3+15 x^2+87 x+43$
• $y^2=19 x^6+48 x^5+88 x^4+90 x^3+45 x+48$
• $y^2=79 x^6+11 x^5+95 x^4+56 x^3+20 x^2+63 x+43$
• $y^2=29 x^6+86 x^5+46 x^4+14 x^3+63 x^2+26 x+74$
• $y^2=2 x^6+49 x^5+3 x^4+16 x^3+59 x^2+76 x+36$
• $y^2=85 x^6+15 x^5+26 x^4+81 x^3+51 x^2+70 x+10$
• $y^2=50 x^6+28 x^5+55 x^4+27 x^3+95 x^2+60 x$
• $y^2=84 x^6+71 x^5+44 x^4+95 x^3+41 x^2+18 x+8$
• $y^2=2 x^6+36 x^5+20 x^4+88 x^3+34 x^2+72 x+83$
• $y^2=77 x^5+49 x^4+88 x^3+95 x^2+64 x+64$
• $y^2=20 x^6+54 x^5+88 x^4+67 x^3+14 x^2+25 x+22$
• $y^2=7 x^6+x^5+49 x^4+88 x^3+46 x^2+84 x+43$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{6}}$.

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.o $\times$ 1.97.t 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.t : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{97^{6}}$ is 1.832972004929.dfna^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{97^{2}}$
  The base change of $A$ to $\F_{97^{2}}$ is 1.9409.agl $\times$ 1.9409.ac. The endomorphism algebra for each factor is:

  • 1.9409.agl : \(\Q(\sqrt{-3}) \).
  • 1.9409.ac : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{97^{3}}$
  The base change of $A$ to $\F_{97^{3}}$ is 1.912673.abze $\times$ 1.912673.bze. The endomorphism algebra for each factor is:

  • 1.912673.abze : \(\Q(\sqrt{-3}) \).
  • 1.912673.bze : \(\Q(\sqrt{-3}) \).

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.af_acu	$2$	(not in LMFDB)
2.97.f_acu	$2$	(not in LMFDB)
2.97.abh_rs	$3$	(not in LMFDB)
2.97.ay_ld	$3$	(not in LMFDB)
2.97.aj_eu	$3$	(not in LMFDB)
2.97.a_agl	$3$	(not in LMFDB)
2.97.a_ac	$3$	(not in LMFDB)
2.97.a_gn	$3$	(not in LMFDB)
2.97.j_eu	$3$	(not in LMFDB)
2.97.y_ld	$3$	(not in LMFDB)