Properties

Label 2.89.a_cs
Base field $\F_{89}$
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:  $\F_{89}$
Dimension:  $2$
L-polynomial:  $1 + 70 x^{2} + 7921 x^{4}$
Frobenius angles:  $\pm0.314326113319$, $\pm0.685673886681$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{3}, \sqrt{-62})\)
Galois group:  $C_2^2$
Jacobians:  $1272$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7992$ $63872064$ $496979970552$ $3937962102137856$ $31181719940022152952$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $90$ $8062$ $704970$ $62764126$ $5584059450$ $496978650142$ $44231334895530$ $3936588817216318$ $350356403707485210$ $31181719950078122302$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 1272 curves (of which all are hyperelliptic):

  • $y^2=70 x^6+19 x^5+44 x^4+65 x^3+11 x^2+41 x+13$
  • $y^2=32 x^6+57 x^5+43 x^4+17 x^3+33 x^2+34 x+39$
  • $y^2=61 x^6+33 x^5+41 x^4+56 x^3+68 x^2+31 x+4$
  • $y^2=50 x^5+56 x^4+70 x^3+74 x^2+48 x+34$
  • $y^2=61 x^5+79 x^4+32 x^3+44 x^2+55 x+13$
  • $y^2=36 x^6+18 x^5+75 x^4+76 x^3+33 x^2+73 x+51$
  • $y^2=83 x^6+22 x^5+59 x^4+19 x^3+67 x^2+68 x+22$
  • $y^2=20 x^5+11 x^4+28 x^3+13 x^2+13 x+10$
  • $y^2=60 x^5+33 x^4+84 x^3+39 x^2+39 x+30$
  • $y^2=46 x^6+84 x^5+14 x^4+71 x^3+82 x^2+26 x+14$
  • $y^2=49 x^6+74 x^5+42 x^4+35 x^3+68 x^2+78 x+42$
  • $y^2=21 x^6+39 x^5+87 x^4+81 x^2+7 x+83$
  • $y^2=84 x^6+49 x^5+55 x^4+7 x^3+69 x^2+85 x+21$
  • $y^2=74 x^6+58 x^5+76 x^4+21 x^3+29 x^2+77 x+63$
  • $y^2=40 x^6+6 x^5+41 x^4+48 x^3+45 x^2+4 x+7$
  • $y^2=31 x^6+18 x^5+34 x^4+55 x^3+46 x^2+12 x+21$
  • $y^2=69 x^6+41 x^5+70 x^4+70 x^3+3 x^2+43 x+7$
  • $y^2=29 x^6+34 x^5+32 x^4+32 x^3+9 x^2+40 x+21$
  • $y^2=14 x^6+16 x^5+78 x^4+57 x^3+84 x^2+23 x+17$
  • $y^2=42 x^6+48 x^5+56 x^4+82 x^3+74 x^2+69 x+51$
  • and 1252 more

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{3}, \sqrt{-62})\).
Endomorphism algebra over $\overline{\F}_{89}$
The base change of $A$ to $\F_{89^{2}}$ is 1.7921.cs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-186}) \)$)$

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.89.a_acs$4$(not in LMFDB)
2.89.as_hp$12$(not in LMFDB)
2.89.s_hp$12$(not in LMFDB)