Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 51 x + 1045 x^{2} - 10149 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0810958877955$, $\pm0.182598387842$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.222573.1 |
Galois group: | $D_{4}$ |
Jacobians: | $24$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30447$ | $1548138609$ | $62078532588333$ | $2459391344344332477$ | $97393911071387418585072$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $149$ | $39091$ | $7877387$ | $1568250139$ | $312080349884$ | $62103854990095$ | $12358664468771189$ | $2459374193351624035$ | $489415464129052796951$ | $97393677359679137558686$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=117x^6+88x^5+130x^4+41x^3+32x^2+167x+101$
- $y^2=88x^6+9x^5+49x^4+155x^3+106x^2+65x+86$
- $y^2=67x^6+47x^5+176x^4+67x^3+99x^2+188x+146$
- $y^2=89x^6+85x^5+123x^4+184x^3+160x^2+70x+141$
- $y^2=56x^6+109x^5+155x^4+81x^3+184x^2+92x+25$
- $y^2=105x^6+30x^5+173x^4+169x^3+71x^2+132x+65$
- $y^2=99x^6+149x^5+57x^4+60x^3+171x^2+10x+70$
- $y^2=54x^6+78x^5+15x^4+111x^3+28x^2+145x+101$
- $y^2=83x^6+32x^5+19x^4+117x^3+27x^2+133x+128$
- $y^2=15x^6+9x^5+77x^4+179x^3+129x^2+145x+143$
- $y^2=140x^6+74x^5+75x^4+35x^3+155x^2+38x+54$
- $y^2=16x^6+2x^5+11x^4+118x^3+66x^2+5x+124$
- $y^2=156x^6+112x^5+187x^4+135x^3+162x^2+x+140$
- $y^2=21x^6+99x^5+84x^4+133x^3+35x^2+x+173$
- $y^2=189x^6+123x^5+197x^4+177x^3+66x^2+162x+178$
- $y^2=193x^6+155x^5+58x^4+146x^3+109x^2+135x+171$
- $y^2=95x^6+137x^5+65x^4+131x^3+89x^2+151x+107$
- $y^2=69x^6+195x^5+101x^4+71x^3+7x^2+146x+121$
- $y^2=163x^6+23x^5+34x^4+68x^3+114x^2+113x+17$
- $y^2=5x^6+13x^5+43x^4+57x^3+47x^2+166x+84$
- $y^2=45x^6+49x^5+93x^4+20x^3+3x^2+138x+34$
- $y^2=148x^6+27x^5+73x^4+157x^3+105x^2+3x+175$
- $y^2=103x^6+91x^5+39x^4+195x^3+160x^2+94x+183$
- $y^2=2x^6+2x^5+112x^4+171x^3+68x^2+85x+185$
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 4.0.222573.1. |
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.bz_bof | $2$ | (not in LMFDB) |