Properties

Label 2.89.a_fm
Base field $\F_{89}$
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:  $\F_{89}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 89 x^{2} )( 1 + 6 x + 89 x^{2} )$
  $1 + 142 x^{2} + 7921 x^{4}$
Frobenius angles:  $\pm0.396989011311$, $\pm0.603010988689$
Angle rank:  $1$ (numerical)
Jacobians:  $1216$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8064$ $65028096$ $496980779904$ $3936046605926400$ $31181719918847992704$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $90$ $8206$ $704970$ $62733598$ $5584059450$ $496980268846$ $44231334895530$ $3936589019311678$ $350356403707485210$ $31181719907729801806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 1216 curves (of which all are hyperelliptic):

  • $y^2=66 x^6+57 x^4+31 x^3+71 x^2+23 x+34$
  • $y^2=20 x^6+82 x^4+4 x^3+35 x^2+69 x+13$
  • $y^2=83 x^6+58 x^5+44 x^4+37 x^3+52 x^2+55 x+63$
  • $y^2=71 x^6+85 x^5+43 x^4+22 x^3+67 x^2+76 x+11$
  • $y^2=38 x^6+15 x^5+67 x^4+58 x^3+15 x^2+68 x+52$
  • $y^2=25 x^6+45 x^5+23 x^4+85 x^3+45 x^2+26 x+67$
  • $y^2=67 x^6+80 x^5+79 x^4+21 x^3+44 x^2+10 x+51$
  • $y^2=23 x^6+62 x^5+59 x^4+63 x^3+43 x^2+30 x+64$
  • $y^2=69 x^6+29 x^5+79 x^4+60 x^3+18 x^2+37 x+49$
  • $y^2=29 x^6+87 x^5+59 x^4+2 x^3+54 x^2+22 x+58$
  • $y^2=46 x^6+2 x^5+46 x^4+30 x^3+46 x^2+2 x+46$
  • $y^2=49 x^6+6 x^5+49 x^4+x^3+49 x^2+6 x+49$
  • $y^2=84 x^6+3 x^5+31 x^4+23 x^3+26 x^2+71 x+59$
  • $y^2=74 x^6+9 x^5+4 x^4+69 x^3+78 x^2+35 x+88$
  • $y^2=62 x^6+24 x^5+45 x^4+66 x^3+55 x^2+82 x+43$
  • $y^2=8 x^6+72 x^5+46 x^4+20 x^3+76 x^2+68 x+40$
  • $y^2=80 x^6+7 x^5+40 x^4+50 x^3+14 x^2+69 x+51$
  • $y^2=62 x^6+21 x^5+31 x^4+61 x^3+42 x^2+29 x+64$
  • $y^2=60 x^6+66 x^5+86 x^4+24 x^3+51 x^2+28 x+77$
  • $y^2=2 x^6+20 x^5+80 x^4+72 x^3+64 x^2+84 x+53$
  • and 1196 more

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89^{2}}$.

Endomorphism algebra over $\F_{89}$
The isogeny class factors as 1.89.ag $\times$ 1.89.g 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 $\overline{\F}_{89}$
The base change of $A$ to $\F_{89^{2}}$ is 1.7921.fm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.89.am_ig$2$(not in LMFDB)
2.89.m_ig$2$(not in LMFDB)
2.89.a_afm$4$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.89.am_ig$2$(not in LMFDB)
2.89.m_ig$2$(not in LMFDB)
2.89.a_afm$4$(not in LMFDB)
2.89.ag_acb$6$(not in LMFDB)
2.89.g_acb$6$(not in LMFDB)