Invariants
| Base field: | $\F_{197}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 53 x + 1095 x^{2} - 10441 x^{3} + 38809 x^{4}$ |
| Frobenius angles: | $\pm0.0572814983224$, $\pm0.140471700629$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.90725.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $29411$ | $1482284989$ | $58405151455991$ | $2268387809378515541$ | $88036388229550573403536$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $145$ | $38191$ | $7639279$ | $1506095115$ | $296709250230$ | $58451734562887$ | $11514990625521703$ | $2268453126342444819$ | $446885265448541491873$ | $88036397287645810315686$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=139 x^6+100 x^5+145 x^4+172 x^3+94 x^2+54 x+143$
- $y^2=48 x^6+68 x^5+110 x^4+93 x^3+55 x^2+76 x+10$
- $y^2=96 x^6+78 x^5+187 x^4+186 x^3+194 x^2+113 x+56$
- $y^2=27 x^6+45 x^5+185 x^4+139 x^3+72 x^2+163 x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{197}$.
Endomorphism algebra over $\F_{197}$| The endomorphism algebra of this simple isogeny class is 4.0.90725.1. |
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 |
|---|---|---|
| 2.197.cb_bqd | $2$ | (not in LMFDB) |