Properties

Label 2.97.a_aes
Base field $\F_{97}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
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:  $1 - 122 x^{2} + 9409 x^{4}$
Frobenius angles:  $\pm0.141759705500$, $\pm0.858240294500$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2}, \sqrt{79})\)
Galois group:  $C_2^2$
Jacobians:  $216$
Cyclic group of points:    no
Non-cyclic primes:   $2, 3$

This isogeny class is simple but not geometrically 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9288$ $86266944$ $832973632776$ $7838130335302656$ $73742412693889452168$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $98$ $9166$ $912674$ $88537150$ $8587340258$ $832975260622$ $80798284478114$ $7837433917541374$ $760231058654565218$ $73742412698286078286$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 216 curves (of which all are hyperelliptic):

  • $y^2=88 x^6+x^5+5 x^4+40 x^3+54 x^2+7 x+10$
  • $y^2=52 x^6+5 x^5+25 x^4+6 x^3+76 x^2+35 x+50$
  • $y^2=76 x^6+59 x^5+66 x^4+83 x^3+79 x^2+54 x+46$
  • $y^2=89 x^6+4 x^5+39 x^4+27 x^3+7 x^2+76 x+36$
  • $y^2=62 x^6+16 x^5+71 x^4+53 x^3+66 x^2+x+16$
  • $y^2=19 x^6+80 x^5+64 x^4+71 x^3+39 x^2+5 x+80$
  • $y^2=61 x^6+24 x^5+47 x^4+38 x^3+42 x^2+36 x+37$
  • $y^2=41 x^6+78 x^4+65 x^3+64 x^2+94 x+77$
  • $y^2=11 x^6+2 x^4+34 x^3+29 x^2+82 x+94$
  • $y^2=65 x^6+8 x^5+24 x^4+19 x^3+51 x^2+24 x+87$
  • $y^2=27 x^6+x^5+21 x^4+60 x^3+60 x^2+68 x+51$
  • $y^2=38 x^6+5 x^5+8 x^4+9 x^3+9 x^2+49 x+61$
  • $y^2=51 x^6+40 x^5+70 x^4+2 x^3+4 x^2+94 x+47$
  • $y^2=61 x^6+6 x^5+59 x^4+10 x^3+20 x^2+82 x+41$
  • $y^2=65 x^6+58 x^5+7 x^4+13 x^3+71 x^2+15 x+33$
  • $y^2=34 x^6+96 x^5+35 x^4+65 x^3+64 x^2+75 x+68$
  • $y^2=40 x^6+53 x^5+59 x^4+89 x^3+31 x^2+67 x+35$
  • $y^2=6 x^6+71 x^5+4 x^4+57 x^3+58 x^2+44 x+78$
  • $y^2=67 x^6+65 x^5+82 x^4+15 x^3+31 x^2+75 x+79$
  • $y^2=37 x^6+25 x^5+24 x^4+69 x^3+57 x^2+82 x+22$
  • and 196 more

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{79})\).
Endomorphism algebra over $\overline{\F}_{97}$
The base change of $A$ to $\F_{97^{2}}$ is 1.9409.aes 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-158}) \)$)$

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.97.a_es$4$(not in LMFDB)
2.97.am_cu$8$(not in LMFDB)
2.97.m_cu$8$(not in LMFDB)