Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 985 x^{2} - 9751 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0229683311136$, $\pm0.235127646112$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4193837.1 |
Galois group: | $D_{4}$ |
Jacobians: | $7$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30787$ | $1551264569$ | $62087233421233$ | $2459377163730601517$ | $97393648060043324074672$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $151$ | $39171$ | $7878493$ | $1568241099$ | $312079507116$ | $62103830652615$ | $12358663986210931$ | $2459374186004953635$ | $489415464042499331029$ | $97393677358945312713886$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=156x^6+169x^5+12x^4+110x^3+15x^2+73x+68$
- $y^2=189x^6+192x^5+109x^4+80x^3+102x^2+38x+140$
- $y^2=84x^6+137x^5+87x^4+172x^3+182x^2+81x+75$
- $y^2=173x^6+90x^5+182x^4+171x^3+13x^2+97x+195$
- $y^2=45x^6+74x^5+165x^4+161x^3+95x^2+19x+150$
- $y^2=101x^6+36x^5+112x^4+4x^3+167x^2+193x+10$
- $y^2=68x^6+146x^5+95x^4+20x^3+101x^2+105x+162$
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 4.0.4193837.1. |
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_blx | $2$ | (not in LMFDB) |