Invariants
| Base field: | $\F_{199}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 28 x + 199 x^{2} )( 1 - 26 x + 199 x^{2} )$ |
| $1 - 54 x + 1126 x^{2} - 10746 x^{3} + 39601 x^{4}$ | |
| Frobenius angles: | $\pm0.0391815390403$, $\pm0.126927281034$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $29928$ | $1542129984$ | $62046417540744$ | $2459267748751656960$ | $97393535783741510069448$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $146$ | $38938$ | $7873310$ | $1568171326$ | $312079147346$ | $62103840486586$ | $12358664343211934$ | $2459374192951194046$ | $489415464139205024210$ | $97393677359900800497178$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=143x^6+9x^5+198x^4+189x^3+156x^2+124x+34$
- $y^2=8x^6+187x^5+46x^4+158x^3+177x^2+115x+103$
- $y^2=2x^6+196x^5+142x^4+70x^3+142x^2+196x+2$
- $y^2=85x^6+119x^5+100x^4+14x^3+26x^2+192x+138$
- $y^2=73x^6+96x^5+171x^4+105x^3+171x^2+96x+73$
- $y^2=16x^6+9x^5+75x^4+188x^3+75x^2+9x+16$
- $y^2=15x^6+17x^5+170x^4+198x^3+12x^2+134x+186$
- $y^2=109x^6+8x^5+180x^4+156x^3+180x^2+8x+109$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$| The isogeny class factors as 1.199.abc $\times$ 1.199.aba 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.