Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 26 x + 193 x^{2} )( 1 - 24 x + 193 x^{2} )$ |
$1 - 50 x + 1010 x^{2} - 9650 x^{3} + 37249 x^{4}$ | |
Frobenius angles: | $\pm0.114714697559$, $\pm0.168091317575$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $36$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28560$ | $1369737600$ | $51664941553680$ | $1925162909429760000$ | $71709211421119313302800$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $144$ | $36770$ | $7186608$ | $1387516798$ | $267786328944$ | $51682562939810$ | $9974730658477008$ | $1925122956902323198$ | $371548729950161169744$ | $71708904873470024818850$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=41x^6+168x^5+172x^4+82x^3+151x^2+93x+135$
- $y^2=88x^6+98x^5+159x^4+169x^3+44x^2+92x+160$
- $y^2=166x^6+167x^5+71x^4+125x^3+71x^2+167x+166$
- $y^2=97x^6+183x^5+53x^4+32x^3+53x^2+183x+97$
- $y^2=90x^6+90x^5+136x^4+2x^3+136x^2+90x+90$
- $y^2=101x^6+6x^5+12x^4+42x^3+12x^2+6x+101$
- $y^2=116x^6+141x^5+57x^4+188x^3+57x^2+141x+116$
- $y^2=94x^6+93x^5+12x^4+95x^3+144x^2+75x+119$
- $y^2=45x^6+184x^5+119x^4+127x^3+111x^2+28x+17$
- $y^2=151x^6+160x^5+50x^4+89x^3+50x^2+160x+151$
- $y^2=37x^6+176x^5+68x^4+162x^3+68x^2+176x+37$
- $y^2=96x^6+81x^5+83x^4+6x^3+83x^2+81x+96$
- $y^2=155x^5+41x^4+46x^3+41x^2+155x$
- $y^2=79x^6+144x^5+12x^4+11x^3+145x^2+181x+155$
- $y^2=164x^6+106x^5+50x^4+142x^3+138x^2+163x+11$
- $y^2=186x^6+101x^5+160x^4+61x^3+160x^2+101x+186$
- $y^2=13x^6+151x^5+99x^4+117x^3+99x^2+151x+13$
- $y^2=151x^6+125x^5+29x^4+28x^3+29x^2+125x+151$
- $y^2=160x^6+98x^5+53x^4+95x^3+141x^2+83x+182$
- $y^2=177x^6+116x^5+140x^4+148x^3+140x^2+116x+177$
- and 16 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$The isogeny class factors as 1.193.aba $\times$ 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.