Invariants
Base field: | $\F_{43}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 43 x^{2} )( 1 + 4 x + 43 x^{2} )$ |
$1 - 4 x + 54 x^{2} - 172 x^{3} + 1849 x^{4}$ | |
Frobenius angles: | $\pm0.291171725172$, $\pm0.598655510457$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $288$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1728$ | $3594240$ | $6326693568$ | $11695081881600$ | $21615539533587648$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $40$ | $1942$ | $79576$ | $3420814$ | $147036040$ | $6321206374$ | $271816714552$ | $11688202178206$ | $502592643980008$ | $21611482302310582$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 288 curves (of which all are hyperelliptic):
- $y^2=18 x^6+37 x^5+39 x^4+5 x^2+21 x+22$
- $y^2=4 x^6+5 x^5+4 x^4+18 x^3+36 x^2+19 x$
- $y^2=32 x^6+33 x^5+20 x^4+9 x^3+37 x^2+12 x+27$
- $y^2=11 x^6+24 x^5+21 x^4+14 x^3+36 x^2+2 x+22$
- $y^2=2 x^6+30 x^5+41 x^4+9 x^3+19 x^2+38 x+24$
- $y^2=11 x^6+31 x^5+31 x^4+9 x^3+12 x^2+24 x+5$
- $y^2=29 x^6+7 x^5+x^4+33 x^3+38 x^2+42 x+5$
- $y^2=4 x^6+14 x^5+19 x^4+21 x^3+37 x^2+27 x+38$
- $y^2=37 x^6+22 x^5+28 x^4+42 x^3+31 x^2+28 x+1$
- $y^2=6 x^6+24 x^5+3 x^4+22 x^3+9 x^2+41 x+28$
- $y^2=6 x^6+23 x^5+38 x^4+17 x^3+40 x^2+16 x+3$
- $y^2=12 x^5+14 x^4+37 x^3+14 x^2+12 x$
- $y^2=18 x^6+6 x^5+16 x^4+25 x^3+14 x^2+5 x+5$
- $y^2=30 x^6+41 x^5+41 x^4+25 x^2+10 x+26$
- $y^2=24 x^5+32 x^4+33 x^3+23 x^2+39 x+19$
- $y^2=9 x^6+36 x^5+37 x^4+15 x^3+39 x^2+33 x+3$
- $y^2=3 x^6+27 x^5+12 x^4+39 x^3+12 x^2+27 x+3$
- $y^2=3 x^6+41 x^5+14 x^4+32 x^3+41 x^2+41$
- $y^2=19 x^6+23 x^5+4 x^4+29 x^3+13 x^2+39 x+33$
- $y^2=25 x^6+20 x^5+39 x^4+27 x^3+39 x^2+20 x+25$
- and 268 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{43}$.
Endomorphism algebra over $\F_{43}$The isogeny class factors as 1.43.ai $\times$ 1.43.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.