Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 51 x + 1035 x^{2} - 9843 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.0925765054042$, $\pm0.159192387568$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.281725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
This isogeny class is simple and 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28391$ | $1367849989$ | $51655058081759$ | $1925125054047145621$ | $71709094761259877212016$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $143$ | $36719$ | $7185233$ | $1387489515$ | $267785893298$ | $51682557277343$ | $9974730602585045$ | $1925122956634332019$ | $371548729954826443019$ | $71708904873647483475614$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=59x^6+162x^5+134x^4+132x^3+114x^2+43x+104$
- $y^2=54x^6+66x^5+118x^4+118x^3+35x^2+64x+130$
- $y^2=38x^6+192x^5+69x^4+49x^3+95x^2+16x+187$
- $y^2=131x^6+91x^5+127x^4+27x^3+11x^2+113x+6$
- $y^2=37x^6+168x^5+148x^4+120x^3+37x^2+133x+129$
- $y^2=132x^6+55x^5+148x^4+105x^3+150x^2+33x+46$
- $y^2=68x^6+102x^5+13x^4+121x^3+113x^2+129x+57$
- $y^2=171x^6+129x^5+187x^4+143x^3+80x^2+111x+36$
- $y^2=77x^6+58x^5+137x^4+48x^3+34x^2+63x+101$
- $y^2=132x^6+154x^5+121x^4+116x^3+122x^2+17x+80$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$The endomorphism algebra of this simple isogeny class is 4.0.281725.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.193.bz_bnv | $2$ | (not in LMFDB) |