Invariants
Base field: | $\F_{163}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 45 x + 831 x^{2} - 7335 x^{3} + 26569 x^{4}$ |
Frobenius angles: | $\pm0.124103562334$, $\pm0.184084818071$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.458725.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20021$ | $696350401$ | $18751284577199$ | $498336354865078621$ | $13239740456683763216336$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $119$ | $26207$ | $4329803$ | $705947091$ | $115064525144$ | $18755383797083$ | $3057125410688573$ | $498311415850618483$ | $81224760542153565269$ | $13239635966983445446022$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=35x^6+115x^5+44x^4+121x^3+72x^2+107x+18$
- $y^2=45x^6+116x^5+14x^4+152x^3+79x^2+124x+140$
- $y^2=33x^6+141x^5+81x^4+16x^3+76x^2+152x+26$
- $y^2=36x^6+129x^5+3x^4+113x^3+80x^2+91x+106$
- $y^2=9x^6+150x^5+21x^4+116x^3+82x^2+52x+13$
- $y^2=112x^6+72x^5+159x^4+109x^3+81x^2+135x+146$
- $y^2=19x^6+110x^5+38x^4+x^3+44x^2+34x+34$
- $y^2=20x^6+87x^5+66x^4+75x^3+12x^2+87x+92$
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 4.0.458725.1. |
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.bt_bfz | $2$ | (not in LMFDB) |