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Properties

Label 2.47.a_aj
Base field $\F_{47}$
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:  $\F_{47}$
Dimension:  $2$
L-polynomial:  $1 - 9 x^{2} + 2209 x^{4}$
Frobenius angles:  $\pm0.234738382599$, $\pm0.765261617401$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-85}, \sqrt{103})\)
Galois group:  $C_2^2$
Jacobians:  $30$
Isomorphism classes:  40
Cyclic group of points:    yes

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2201$ $4844401$ $10779274244$ $23853641592361$ $52599132024237161$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $48$ $2192$ $103824$ $4888356$ $229345008$ $10779333158$ $506623120464$ $23811268561348$ $1119130473102768$ $52599131812644272$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

  • $y^2=35 x^6+19 x^5+17 x^4+11 x^3+14 x^2+7 x+39$
  • $y^2=40 x^6+13 x^5+45 x^4+31 x^3+37 x^2+40 x+36$
  • $y^2=28 x^6+13 x^5+29 x^4+18 x^3+32 x^2+34 x+36$
  • $y^2=46 x^6+18 x^5+4 x^4+43 x^3+19 x^2+29 x+39$
  • $y^2=25 x^6+3 x^5+32 x^4+17 x^3+23 x^2+31 x+35$
  • $y^2=31 x^6+15 x^5+19 x^4+38 x^3+21 x^2+14 x+34$
  • $y^2=5 x^6+3 x^5+37 x^4+46 x^3+20 x^2+13 x+21$
  • $y^2=23 x^6+6 x^5+10 x^4+x^3+35 x^2+45 x+10$
  • $y^2=21 x^6+30 x^5+3 x^4+5 x^3+34 x^2+37 x+3$
  • $y^2=13 x^6+34 x^5+10 x^4+21 x^3+27 x^2+45 x+15$
  • $y^2=18 x^6+29 x^5+3 x^4+11 x^3+41 x^2+37 x+28$
  • $y^2=10 x^6+8 x^5+39 x^4+36 x^3+34 x^2+22 x+21$
  • $y^2=19 x^6+39 x^5+20 x^4+12 x^3+35 x^2+11 x+31$
  • $y^2=x^6+7 x^5+6 x^4+13 x^3+34 x^2+8 x+14$
  • $y^2=39 x^6+6 x^5+13 x^4+45 x^3+x^2+24 x+3$
  • $y^2=7 x^6+30 x^5+18 x^4+37 x^3+5 x^2+26 x+15$
  • $y^2=18 x^6+17 x^5+22 x^4+13 x^3+38 x^2+35 x+24$
  • $y^2=7 x^6+20 x^5+25 x^4+45 x^3+11 x^2+8 x+23$
  • $y^2=35 x^6+6 x^5+31 x^4+37 x^3+8 x^2+40 x+21$
  • $y^2=27 x^6+8 x^5+25 x^4+7 x^3+41 x^2+37 x+20$
  • $y^2=15 x^6+29 x^5+38 x^4+24 x^3+9 x^2+14 x+45$
  • $y^2=12 x^6+14 x^5+40 x^3+39 x^2+11 x+29$
  • $y^2=13 x^6+23 x^5+12 x^3+7 x^2+8 x+4$
  • $y^2=43 x^6+41 x^5+39 x^4+7 x^3+38 x^2+29 x+15$
  • $y^2=22 x^6+33 x^5+32 x^4+23 x^3+18 x^2+8 x+26$
  • $y^2=16 x^6+24 x^5+19 x^4+21 x^3+43 x^2+40 x+36$
  • $y^2=10 x^6+22 x^5+15 x^4+23 x^3+42 x^2+43 x+19$
  • $y^2=6 x^6+17 x^5+27 x^4+43 x^3+40 x^2+44 x+18$
  • $y^2=2 x^6+12 x^5+3 x^4+21 x^3+19 x^2+13 x+40$
  • $y^2=10 x^6+13 x^5+15 x^4+11 x^3+x^2+18 x+12$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{2}}$.

Endomorphism algebra over $\F_{47}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-85}, \sqrt{103})\).
Endomorphism algebra over $\overline{\F}_{47}$
The base change of $A$ to $\F_{47^{2}}$ is 1.2209.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-8755}) \)$)$

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.47.a_j$4$(not in LMFDB)