Properties

Label 2.79.a_cg
Base field $\F_{79}$
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:  $\F_{79}$
Dimension:  $2$
L-polynomial:  $( 1 - 10 x + 79 x^{2} )( 1 + 10 x + 79 x^{2} )$
  $1 + 58 x^{2} + 6241 x^{4}$
Frobenius angles:  $\pm0.309822710654$, $\pm0.690177289346$
Angle rank:  $1$ (numerical)
Jacobians:  $1088$
Cyclic group of points:    no
Non-cyclic primes:   $2, 3, 5$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6300$ $39690000$ $243086564700$ $1517819264640000$ $9468276088490257500$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $80$ $6358$ $493040$ $38968318$ $3077056400$ $243085673878$ $19203908986160$ $1517108799431038$ $119851595982618320$ $9468276094353667798$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 1088 curves (of which all are hyperelliptic):

  • $y^2=11 x^6+50 x^5+45 x^4+66 x^3+75 x^2+40 x+31$
  • $y^2=33 x^6+71 x^5+56 x^4+40 x^3+67 x^2+41 x+14$
  • $y^2=75 x^6+4 x^5+62 x^4+52 x^3+37 x^2+43 x+1$
  • $y^2=67 x^6+12 x^5+28 x^4+77 x^3+32 x^2+50 x+3$
  • $y^2=46 x^6+31 x^5+40 x^4+24 x^3+52 x^2+39 x+37$
  • $y^2=59 x^6+14 x^5+41 x^4+72 x^3+77 x^2+38 x+32$
  • $y^2=68 x^6+14 x^5+72 x^4+11 x^3+57 x^2+20 x+58$
  • $y^2=46 x^6+42 x^5+58 x^4+33 x^3+13 x^2+60 x+16$
  • $y^2=69 x^6+78 x^5+67 x^4+x^3+52 x^2+55 x+30$
  • $y^2=49 x^6+76 x^5+43 x^4+3 x^3+77 x^2+7 x+11$
  • $y^2=40 x^6+61 x^5+12 x^4+32 x^3+27 x^2+8 x+39$
  • $y^2=41 x^6+25 x^5+36 x^4+17 x^3+2 x^2+24 x+38$
  • $y^2=54 x^6+10 x^5+29 x^4+24 x^3+39 x^2+5 x$
  • $y^2=4 x^6+30 x^5+8 x^4+72 x^3+38 x^2+15 x$
  • $y^2=39 x^6+39 x^5+36 x^4+78 x^3+41 x^2+15 x+61$
  • $y^2=38 x^6+38 x^5+29 x^4+76 x^3+44 x^2+45 x+25$
  • $y^2=7 x^6+75 x^5+17 x^4+26 x^3+4 x^2+17 x+72$
  • $y^2=68 x^6+27 x^5+50 x^4+47 x^3+37 x^2+27 x+45$
  • $y^2=46 x^6+2 x^5+71 x^4+62 x^3+32 x^2+2 x+56$
  • $y^2=41 x^6+23 x^5+48 x^4+24 x^2+37 x+10$
  • and 1068 more

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{2}}$.

Endomorphism algebra over $\F_{79}$
The isogeny class factors as 1.79.ak $\times$ 1.79.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 $\overline{\F}_{79}$
The base change of $A$ to $\F_{79^{2}}$ is 1.6241.cg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.79.au_jy$2$(not in LMFDB)
2.79.u_jy$2$(not in LMFDB)
2.79.a_acg$4$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.79.au_jy$2$(not in LMFDB)
2.79.u_jy$2$(not in LMFDB)
2.79.a_acg$4$(not in LMFDB)
2.79.ak_v$6$(not in LMFDB)
2.79.k_v$6$(not in LMFDB)