Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 25 x + 199 x^{2} )^{2}$ |
$1 - 50 x + 1023 x^{2} - 9950 x^{3} + 39601 x^{4}$ | |
Frobenius angles: | $\pm0.153403448314$, $\pm0.153403448314$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $47$ |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30625$ | $1550390625$ | $62092824010000$ | $2459460991222265625$ | $97394196115326732015625$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $150$ | $39148$ | $7879200$ | $1568294548$ | $312081263250$ | $62103871141198$ | $12358664711933550$ | $2459374196294485348$ | $489415464151483077600$ | $97393677359561822670748$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 47 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=97x^6+180x^5+130x^4+148x^3+130x^2+180x+97$
- $y^2=153x^6+66x^5+x^4+187x^3+x^2+66x+153$
- $y^2=2x^6+186x^5+126x^4+40x^3+126x^2+186x+2$
- $y^2=38x^6+9x^5+187x^4+78x^3+162x^2+160x+184$
- $y^2=190x^6+57x^5+166x^4+193x^3+166x^2+57x+190$
- $y^2=3x^6+55x^3+156$
- $y^2=32x^6+146x^5+102x^4+18x^3+148x^2+2x+121$
- $y^2=37x^6+197x^5+68x^4+195x^3+65x^2+157x+104$
- $y^2=142x^6+50x^5+16x^4+101x^3+16x^2+50x+142$
- $y^2=147x^6+188x^5+66x^4+66x^3+66x^2+188x+147$
- $y^2=135x^6+51x^5+90x^4+120x^3+116x^2+99x+30$
- $y^2=27x^6+118x^5+19x^4+130x^3+110x^2+3x+75$
- $y^2=138x^6+97x^5+195x^4+43x^3+195x^2+97x+138$
- $y^2=162x^6+11x^5+65x^3+11x+162$
- $y^2=3x^6+3x^3+176$
- $y^2=115x^6+117x^5+179x^4+186x^3+179x^2+117x+115$
- $y^2=48x^6+107x^5+34x^4+79x^3+34x^2+107x+48$
- $y^2=120x^6+128x^5+150x^4+162x^3+193x^2+177x+73$
- $y^2=135x^6+88x^5+158x^4+22x^3+69x^2+40x+19$
- $y^2=173x^6+175x^5+91x^4+8x^3+34x^2+22x+39$
- and 27 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$The isogeny class factors as 1.199.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.