Invariants
Base field: | $\F_{163}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 46 x + 853 x^{2} - 7498 x^{3} + 26569 x^{4}$ |
Frobenius angles: | $\pm0.0946297427759$, $\pm0.179383193902$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.666176.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $19879$ | $695108993$ | $18746213076100$ | $498321129823562249$ | $13239703393800730092079$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $118$ | $26160$ | $4328632$ | $705925524$ | $115064203038$ | $18755379822270$ | $3057125372006026$ | $498311415624273444$ | $81224760543614590456$ | $13239635967061498988800$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=118x^6+114x^5+17x^4+49x^3+47x^2+29x+129$
- $y^2=22x^6+72x^5+155x^4+118x^3+134x^2+94x+137$
- $y^2=84x^6+69x^5+161x^4+49x^3+130x^2+54x+16$
- $y^2=80x^6+160x^5+139x^4+73x^3+24x^2+37x+114$
- $y^2=147x^6+155x^5+154x^4+36x^3+118x^2+126x+37$
- $y^2=32x^6+16x^5+149x^4+25x^3+48x^2+162x+139$
- $y^2=44x^6+116x^5+10x^4+106x^3+4x^2+80x+28$
- $y^2=79x^6+15x^5+18x^4+129x^3+136x^2+105x+31$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{163}$.
Endomorphism algebra over $\F_{163}$The endomorphism algebra of this simple isogeny class is 4.0.666176.1. |
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 |
---|---|---|
2.163.bu_bgv | $2$ | (not in LMFDB) |