Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 26 x + 199 x^{2} )( 1 - 23 x + 199 x^{2} )$ |
$1 - 49 x + 996 x^{2} - 9751 x^{3} + 39601 x^{4}$ | |
Frobenius angles: | $\pm0.126927281034$, $\pm0.196619630811$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $12$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30798$ | $1552157604$ | $62099991653544$ | $2459474497819712160$ | $97394168674300405771098$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $151$ | $39193$ | $7880110$ | $1568303161$ | $312081175321$ | $62103865456186$ | $12358664572218199$ | $2459374193973310801$ | $489415464125815083010$ | $97393677359488231291153$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=108x^6+198x^5+178x^4+167x^3+102x^2+45x+17$
- $y^2=65x^6+65x^5+4x^4+111x^3+180x^2+20x+124$
- $y^2=182x^6+75x^5+182x^4+136x^3+149x^2+115x+183$
- $y^2=149x^6+178x^5+182x^4+62x^3+54x^2+148x+120$
- $y^2=38x^6+23x^5+14x^4+88x^3+187x^2+137x+30$
- $y^2=12x^6+8x^5+144x^4+98x^3+179x^2+26$
- $y^2=65x^6+168x^5+87x^4+42x^3+145x^2+111x+48$
- $y^2=38x^6+118x^5+17x^4+181x^3+37x^2+114x+196$
- $y^2=x^6+123x^5+29x^4+111x^3+76x^2+136x+35$
- $y^2=81x^6+9x^5+122x^4+127x^3+75x^2+13x+187$
- $y^2=166x^6+185x^5+101x^4+109x^3+29x^2+134x+182$
- $y^2=22x^6+17x^5+113x^4+26x^3+108x^2+50x+77$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$The isogeny class factors as 1.199.aba $\times$ 1.199.ax and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ad_ahs | $2$ | (not in LMFDB) |
2.199.d_ahs | $2$ | (not in LMFDB) |
2.199.bx_bmi | $2$ | (not in LMFDB) |