Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 52 x + 1071 x^{2} - 10348 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0588869325648$, $\pm0.170366934480$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.803088.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30273$ | $1546132929$ | $62067793800276$ | $2459349504967905033$ | $97393779093852815039313$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $148$ | $39040$ | $7876024$ | $1568223460$ | $312079926988$ | $62103849313174$ | $12358664401491556$ | $2459374192577916868$ | $489415464118611355624$ | $97393677359494378528480$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=64x^6+86x^5+117x^4+189x^3+55x^2+67x+148$
- $y^2=73x^6+47x^5+109x^4+92x^3+39x^2+13x+169$
- $y^2=166x^6+109x^5+135x^4+22x^3+26x^2+36x+94$
- $y^2=34x^6+27x^5+89x^4+8x^3+21x^2+3x+88$
- $y^2=129x^6+169x^5+148x^4+143x^3+69x^2+196x+195$
- $y^2=99x^6+162x^5+134x^4+11x^3+176x^2+103x+10$
- $y^2=72x^6+172x^5+145x^4+3x^3+41x^2+61x+79$
- $y^2=46x^6+42x^5+130x^4+173x^3+21x^2+61x+148$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$The endomorphism algebra of this simple isogeny class is 4.0.803088.1. |
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 |
---|---|---|
2.199.ca_bpf | $2$ | (not in LMFDB) |