Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 27 x + 193 x^{2} )( 1 - 24 x + 193 x^{2} )$ |
$1 - 51 x + 1034 x^{2} - 9843 x^{3} + 37249 x^{4}$ | |
Frobenius angles: | $\pm0.0758389534121$, $\pm0.168091317575$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $9$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28390$ | $1367773420$ | $51653956563040$ | $1925116359306566400$ | $71709045254225812448950$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $143$ | $36717$ | $7185080$ | $1387483249$ | $267785708423$ | $51682552897794$ | $9974730515120759$ | $1925122955132512801$ | $371548729932726822680$ | $71708904873379222119357$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=17x^6+161x^5+13x^4+188x^3+174x^2+10x+71$
- $y^2=102x^6+123x^5+54x^4+59x^3+124x^2+37x+142$
- $y^2=94x^6+23x^5+102x^4+90x^3+42x^2+117x+98$
- $y^2=61x^6+60x^5+50x^4+29x^3+130x^2+72x+90$
- $y^2=125x^6+188x^5+152x^4+86x^3+168x^2+30x+62$
- $y^2=96x^6+165x^5+151x^4+99x^3+165x^2+75x+120$
- $y^2=74x^6+115x^5+159x^4+183x^3+117x^2+46x+115$
- $y^2=178x^6+188x^5+60x^4+171x^3+10x^2+110x+4$
- $y^2=64x^6+77x^5+71x^4+103x^3+111x^2+14x+49$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$The isogeny class factors as 1.193.abb $\times$ 1.193.ay 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.