Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 50 x + 1016 x^{2} - 9950 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0639639324574$, $\pm0.208870266773$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.90944.1 |
Galois group: | $D_{4}$ |
Jacobians: | $36$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30618$ | $1549821924$ | $62084538832050$ | $2459395976883917904$ | $97393835662033816874178$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $150$ | $39134$ | $7878150$ | $1568253094$ | $312080108250$ | $62103846002798$ | $12358664269243050$ | $2459374190063741374$ | $489415464088360906950$ | $97393677359360074887854$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=135x^6+40x^5+198x^4+87x^3+196x^2+119x+184$
- $y^2=84x^6+130x^5+103x^4+195x^3+97x^2+182x+143$
- $y^2=25x^6+144x^5+166x^4+146x^3+21x^2+162x+70$
- $y^2=164x^6+47x^5+22x^4+63x^3+164x^2+108x+97$
- $y^2=123x^6+86x^5+174x^4+38x^3+133x^2+116x+153$
- $y^2=44x^6+22x^5+14x^4+56x^3+108x^2+23x+175$
- $y^2=161x^6+154x^5+13x^4+53x^3+189x^2+19x+39$
- $y^2=169x^6+175x^5+76x^4+138x^3+4x^2+84x+144$
- $y^2=37x^6+6x^5+148x^4+128x^3+46x^2+159x+195$
- $y^2=63x^6+65x^5+54x^4+65x^3+35x^2+88x+25$
- $y^2=111x^6+67x^5+177x^4+57x^3+145x^2+131x+79$
- $y^2=42x^6+105x^5+180x^4+150x^3+103x^2+36x+88$
- $y^2=35x^6+152x^5+165x^4+193x^3+113x^2+50x+111$
- $y^2=143x^6+16x^5+37x^4+166x^3+169x^2+114x+16$
- $y^2=152x^6+120x^5+16x^4+178x^3+24x^2+57x+112$
- $y^2=43x^6+14x^5+173x^4+132x^3+154x^2+175x+97$
- $y^2=3x^6+109x^5+155x^4+49x^3+50x^2+131x+149$
- $y^2=121x^6+195x^5+87x^4+184x^3+138x^2+184x+32$
- $y^2=50x^6+123x^5+122x^4+87x^3+59x^2+182x+1$
- $y^2=161x^6+121x^5+12x^4+41x^3+140x^2+191x+30$
- and 16 more
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 4.0.90944.1. |
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.by_bnc | $2$ | (not in LMFDB) |