Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 24 x + 193 x^{2} )^{2}$ |
$1 - 48 x + 962 x^{2} - 9264 x^{3} + 37249 x^{4}$ | |
Frobenius angles: | $\pm0.168091317575$, $\pm0.168091317575$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $34$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28900$ | $1373443600$ | $51683590156900$ | $1925229510696960000$ | $71709390988296636422500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $146$ | $36870$ | $7189202$ | $1387564798$ | $267786999506$ | $51682569295110$ | $9974730665550482$ | $1925122955520115198$ | $371548729910257442066$ | $71708904872702398969350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 34 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=76x^6+161x^5+28x^4+118x^3+28x^2+161x+76$
- $y^2=139x^6+191x^5+137x^4+79x^3+75x^2+31x+6$
- $y^2=27x^6+126x^5+148x^4+16x^3+67x^2+54x+26$
- $y^2=158x^6+108x^5+67x^4+56x^3+23x^2+85x+133$
- $y^2=172x^6+99x^5+160x^4+191x^3+174x^2+103x+137$
- $y^2=180x^6+46x^5+43x^4+88x^3+90x^2+148x+152$
- $y^2=123x^6+172x^5+149x^4+40x^3+61x^2+4x+40$
- $y^2=182x^6+6x^5+2x^4+125x^3+100x^2+24x+157$
- $y^2=187x^6+100x^5+112x^4+44x^3+89x^2+134x+109$
- $y^2=15x^5+143x^4+135x^3+98x^2+141x$
- $y^2=72x^6+84x^5+190x^4+96x^3+98x^2+190x$
- $y^2=91x^6+144x^5+68x^4+38x^3+106x^2+3x+82$
- $y^2=149x^6+97x^5+59x^4+82x^3+101x^2+191x+141$
- $y^2=129x^6+121x^5+86x^4+48x^3+48x^2+126x+67$
- $y^2=26x^6+90x^5+102x^4+102x^2+90x+26$
- $y^2=68x^6+165x^5+106x^4+31x^3+174x^2+81x+35$
- $y^2=64x^6+169x^5+30x^4+12x^3+30x^2+169x+64$
- $y^2=36x^6+82x^5+57x^4+168x^3+119x^2+164x+127$
- $y^2=7x^6+154x^5+72x^4+156x^3+145x^2+47x+98$
- $y^2=127x^6+100x^5+67x^4+116x^3+64x^2+115x+29$
- and 14 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.ay 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.