Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 978 x^{2} - 9457 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.0549247165904$, $\pm0.216032287700$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.7540236.1 |
Galois group: | $D_{4}$ |
Jacobians: | $16$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28722$ | $1371015948$ | $51666339579336$ | $1925137335886408704$ | $71708993036323845168882$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $145$ | $36805$ | $7186804$ | $1387498369$ | $267785513425$ | $51682541802370$ | $9974730247419985$ | $1925122950570133249$ | $371548729874712505012$ | $71708904872892405475525$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+58x^5+191x^4+29x^3+156x^2+104x+88$
- $y^2=42x^6+86x^5+131x^4+72x^3+36x^2+61x+78$
- $y^2=34x^6+185x^5+185x^4+101x^3+145x^2+7x+39$
- $y^2=172x^6+107x^5+97x^4+156x^3+150x^2+168x+53$
- $y^2=151x^6+31x^5+91x^4+180x^3+136x^2+191x+2$
- $y^2=164x^6+6x^5+73x^4+128x^3+57x^2+133x+62$
- $y^2=163x^6+172x^5+147x^4+121x^3+112x^2+18x+26$
- $y^2=141x^6+69x^5+100x^4+118x^3+159x^2+97x+141$
- $y^2=44x^6+137x^5+124x^4+118x^3+190x^2+69x+86$
- $y^2=176x^6+x^5+60x^4+161x^3+28x^2+21x+164$
- $y^2=52x^6+169x^5+139x^4+93x^3+35x^2+182x+141$
- $y^2=174x^6+16x^5+91x^4+30x^3+118x^2+70x+136$
- $y^2=128x^6+111x^5+135x^4+76x^3+60x^2+124x+45$
- $y^2=7x^6+13x^5+23x^4+39x^3+60x^2+7x+52$
- $y^2=99x^6+120x^5+157x^4+55x^3+188x^2+2x+190$
- $y^2=121x^6+116x^5+143x^4+113x^3+166x^2+172x+71$
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 4.0.7540236.1. |
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.bx_blq | $2$ | (not in LMFDB) |