Invariants
Base field: | $\F_{163}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 44 x + 799 x^{2} - 7172 x^{3} + 26569 x^{4}$ |
Frobenius angles: | $\pm0.0415917694910$, $\pm0.238728410361$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.6491408.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20153$ | $696991505$ | $18750032701988$ | $498316045500751625$ | $13239638037633233537073$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $120$ | $26232$ | $4329516$ | $705918324$ | $115063635040$ | $18755365271214$ | $3057125117133312$ | $498311412254357796$ | $81224760510823322388$ | $13239635966878278506632$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=41x^6+129x^5+130x^4+159x^3+86x^2+143x+81$
- $y^2=3x^6+96x^5+76x^4+161x^3+36x^2+62x+83$
- $y^2=70x^6+117x^5+93x^4+72x^3+5x^2+13x+13$
- $y^2=86x^6+127x^5+89x^4+19x^3+138x^2+156x+34$
- $y^2=68x^6+31x^5+30x^4+127x^3+127x^2+42x+151$
- $y^2=138x^6+100x^5+38x^4+18x^3+14x^2+105x+96$
- $y^2=150x^6+138x^5+147x^4+39x^3+32x^2+135x+154$
- $y^2=86x^6+49x^5+13x^4+37x^3+114x^2+82x+71$
- $y^2=103x^6+122x^5+130x^4+47x^3+x^2+82x+135$
- $y^2=32x^6+86x^5+150x^4+113x^3+158x^2+122x+80$
- $y^2=98x^6+61x^5+152x^4+48x^3+2x^2+21x+7$
- $y^2=22x^6+57x^5+36x^4+86x^3+35x^2+43x+161$
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 4.0.6491408.1. |
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.bs_bet | $2$ | (not in LMFDB) |