Properties

Label 2.97.a_dq
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

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Invariants

Base field:  $\F_{97}$
Dimension:  $2$
L-polynomial:  $( 1 - 10 x + 97 x^{2} )( 1 + 10 x + 97 x^{2} )$
  $1 + 94 x^{2} + 9409 x^{4}$
Frobenius angles:  $\pm0.330505784077$, $\pm0.669494215923$
Angle rank:  $1$ (numerical)
Jacobians:  $1173$
Cyclic group of points:    no
Non-cyclic primes:   $2, 3$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9504$ $90326016$ $832970182176$ $7839201269661696$ $73742412699365804064$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $98$ $9598$ $912674$ $88549246$ $8587340258$ $832968359422$ $80798284478114$ $7837433749213438$ $760231058654565218$ $73742412709238782078$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 1173 curves (of which all are hyperelliptic):

  • $y^2=15 x^6+94 x^5+67 x^4+90 x^3+93 x^2+36 x+15$
  • $y^2=75 x^6+82 x^5+44 x^4+62 x^3+77 x^2+83 x+75$
  • $y^2=17 x^6+66 x^5+7 x^4+19 x^3+63 x^2+53 x+1$
  • $y^2=85 x^6+39 x^5+35 x^4+95 x^3+24 x^2+71 x+5$
  • $y^2=4 x^6+27 x^5+92 x^4+70 x^3+92 x^2+27 x+4$
  • $y^2=20 x^6+38 x^5+72 x^4+59 x^3+72 x^2+38 x+20$
  • $y^2=81 x^6+56 x^5+20 x^4+x^2+77 x+36$
  • $y^2=17 x^6+86 x^5+3 x^4+5 x^2+94 x+83$
  • $y^2=80 x^6+4 x^5+29 x^4+18 x^3+88 x^2+31 x+61$
  • $y^2=12 x^6+20 x^5+48 x^4+90 x^3+52 x^2+58 x+14$
  • $y^2=86 x^6+15 x^5+24 x^4+93 x^3+41 x^2+42 x+44$
  • $y^2=42 x^6+75 x^5+23 x^4+77 x^3+11 x^2+16 x+26$
  • $y^2=45 x^6+8 x^5+71 x^4+21 x^3+x^2+69 x+77$
  • $y^2=31 x^6+40 x^5+64 x^4+8 x^3+5 x^2+54 x+94$
  • $y^2=66 x^6+72 x^5+64 x^4+30 x^3+37 x^2+36 x+60$
  • $y^2=22 x^6+34 x^5+75 x^4+81 x^3+30 x^2+89 x+21$
  • $y^2=13 x^6+73 x^5+84 x^4+17 x^3+53 x^2+57 x+8$
  • $y^2=33 x^6+51 x^5+4 x^4+57 x^3+46 x^2+6 x+75$
  • $y^2=49 x^6+78 x^5+2 x^4+32 x^3+85 x^2+81 x+69$
  • $y^2=51 x^6+2 x^5+10 x^4+63 x^3+37 x^2+17 x+54$
  • and 1153 more

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.ak $\times$ 1.97.k and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{97}$
The base change of $A$ to $\F_{97^{2}}$ is 1.9409.dq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.97.au_li$2$(not in LMFDB)
2.97.u_li$2$(not in LMFDB)
2.97.a_adq$4$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.97.au_li$2$(not in LMFDB)
2.97.u_li$2$(not in LMFDB)
2.97.a_adq$4$(not in LMFDB)
2.97.ak_d$6$(not in LMFDB)
2.97.k_d$6$(not in LMFDB)
2.97.ay_lc$8$(not in LMFDB)
2.97.y_lc$8$(not in LMFDB)