Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 987 x^{2} - 9751 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0508592358596$, $\pm0.230295663413$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.175725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $32$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30789$ | $1551426921$ | $62089553046411$ | $2459394917225867709$ | $97393744093398804550224$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $151$ | $39175$ | $7878787$ | $1568252419$ | $312079814836$ | $62103837262651$ | $12358664105309449$ | $2459374187888862259$ | $489415464070114958593$ | $97393677359349722595550$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=118x^6+196x^5+195x^4+6x^3+79x^2+3x+33$
- $y^2=183x^6+47x^5+111x^4+6x^3+73x^2+6x+159$
- $y^2=13x^6+178x^5+150x^4+155x^3+33x^2+116x+53$
- $y^2=113x^6+70x^5+49x^4+53x^3+127x^2+41x+107$
- $y^2=106x^6+94x^5+126x^4+194x^3+178x^2+156x+109$
- $y^2=68x^6+169x^5+57x^4+85x^3+146x^2+14x+151$
- $y^2=154x^6+125x^5+130x^4+66x^3+47x^2+197x+129$
- $y^2=192x^6+150x^5+55x^4+94x^3+192x^2+112x+96$
- $y^2=47x^6+88x^5+39x^4+160x^3+45x^2+27x+68$
- $y^2=78x^6+67x^5+22x^4+141x^3+72x^2+70x+148$
- $y^2=159x^6+186x^5+92x^4+163x^3+112x^2+74x+172$
- $y^2=146x^6+75x^5+145x^4+74x^3+160x^2+42x+180$
- $y^2=81x^6+48x^5+139x^4+162x^3+60x^2+63x+197$
- $y^2=137x^6+171x^5+198x^4+77x^3+5x^2+108x+142$
- $y^2=86x^6+3x^5+111x^4+112x^3+75x^2+59x+35$
- $y^2=73x^6+102x^4+162x^3+185x^2+59x+9$
- $y^2=75x^6+152x^5+13x^4+125x^3+176x^2+110x+163$
- $y^2=184x^6+82x^5+192x^4+128x^3+79x^2+113x+194$
- $y^2=74x^6+195x^5+101x^4+7x^3+63x^2+175x+98$
- $y^2=120x^6+87x^5+130x^4+4x^3+44x^2+125x+73$
- and 12 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 4.0.175725.1. |
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.bx_blz | $2$ | (not in LMFDB) |