Properties

Label 2.73.a_eg
Base field $\F_{73}$
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:  $\F_{73}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 73 x^{2} )( 1 + 6 x + 73 x^{2} )$
  $1 + 110 x^{2} + 5329 x^{4}$
Frobenius angles:  $\pm0.385799748780$, $\pm0.614200251220$
Angle rank:  $1$ (numerical)
Jacobians:  $707$
Cyclic group of points:    no
Non-cyclic primes:   $2$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5440$ $29593600$ $151333798720$ $806378250240000$ $4297625825963195200$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $74$ $5550$ $389018$ $28395358$ $2073071594$ $151333371150$ $11047398519098$ $806460201328318$ $58871586708267914$ $4297625822222832750$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 707 curves (of which all are hyperelliptic):

  • $y^2=44 x^6+42 x^5+17 x^4+60 x^3+37 x^2+25 x+69$
  • $y^2=x^6+64 x^5+12 x^4+8 x^3+39 x^2+52 x+53$
  • $y^2=12 x^6+26 x^5+56 x^4+61 x^3+67 x^2+55 x+20$
  • $y^2=60 x^6+57 x^5+61 x^4+13 x^3+43 x^2+56 x+27$
  • $y^2=60 x^6+3 x^5+48 x^4+20 x^3+16 x^2+5 x+26$
  • $y^2=8 x^6+15 x^5+21 x^4+27 x^3+7 x^2+25 x+57$
  • $y^2=23 x^6+68 x^5+33 x^4+41 x^3+33 x^2+68 x+23$
  • $y^2=42 x^6+48 x^5+19 x^4+59 x^3+19 x^2+48 x+42$
  • $y^2=28 x^6+67 x^5+6 x^4+25 x^3+68 x^2+54 x+44$
  • $y^2=29 x^6+6 x^5+34 x^4+37 x^3+12 x^2+44 x+46$
  • $y^2=72 x^6+30 x^5+24 x^4+39 x^3+60 x^2+x+11$
  • $y^2=14 x^6+5 x^5+25 x^3+17 x^2+46 x+51$
  • $y^2=70 x^6+25 x^5+52 x^3+12 x^2+11 x+36$
  • $y^2=43 x^6+4 x^5+x^4+60 x^3+39 x^2+50 x+62$
  • $y^2=69 x^6+20 x^5+5 x^4+8 x^3+49 x^2+31 x+18$
  • $y^2=48 x^6+71 x^5+23 x^4+38 x^3+3 x^2+8 x+59$
  • $y^2=21 x^6+63 x^5+42 x^4+44 x^3+15 x^2+40 x+3$
  • $y^2=38 x^6+2 x^5+11 x^4+21 x^3+18 x^2+21 x+41$
  • $y^2=14 x^5+33 x^4+59 x^3+25 x^2+7 x+43$
  • $y^2=70 x^5+19 x^4+3 x^3+52 x^2+35 x+69$
  • and 687 more

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{2}}$.

Endomorphism algebra over $\F_{73}$
The isogeny class factors as 1.73.ag $\times$ 1.73.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 $\overline{\F}_{73}$
The base change of $A$ to $\F_{73^{2}}$ is 1.5329.eg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.73.am_ha$2$(not in LMFDB)
2.73.m_ha$2$(not in LMFDB)
2.73.abg_pm$4$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.73.am_ha$2$(not in LMFDB)
2.73.m_ha$2$(not in LMFDB)
2.73.abg_pm$4$(not in LMFDB)
2.73.aw_ji$4$(not in LMFDB)
2.73.ak_by$4$(not in LMFDB)
2.73.a_aeg$4$(not in LMFDB)
2.73.k_by$4$(not in LMFDB)
2.73.w_ji$4$(not in LMFDB)
2.73.bg_pm$4$(not in LMFDB)
2.73.ag_abl$6$(not in LMFDB)
2.73.g_abl$6$(not in LMFDB)
2.73.a_ads$8$(not in LMFDB)
2.73.a_ds$8$(not in LMFDB)
2.73.aq_hb$12$(not in LMFDB)
2.73.q_hb$12$(not in LMFDB)