Invariants
| Base field: | $\F_{197}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 48 x + 955 x^{2} - 9456 x^{3} + 38809 x^{4}$ |
| Frobenius angles: | $\pm0.0378626054803$, $\pm0.245516893279$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.15296400.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $30261$ | $1490929209$ | $58440714873156$ | $2268465920779182201$ | $88036373631149825973861$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $150$ | $38416$ | $7643934$ | $1506146980$ | $296709201030$ | $58451718040702$ | $11514990196396158$ | $2268453119207520964$ | $446885265364235686278$ | $88036397287032117448336$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=104 x^6+37 x^5+47 x^4+104 x^3+120 x^2+148 x+20$
- $y^2=12 x^6+76 x^5+54 x^4+5 x^3+162 x^2+55 x+109$
- $y^2=121 x^6+58 x^5+184 x^4+5 x^3+41 x^2+147 x+186$
- $y^2=103 x^6+149 x^5+156 x^4+68 x^3+58 x^2+120 x+31$
- $y^2=48 x^6+63 x^5+167 x^4+144 x^3+28 x^2+141 x+172$
- $y^2=133 x^6+152 x^5+56 x^4+7 x^3+173 x^2+18 x+40$
- $y^2=67 x^6+64 x^5+158 x^4+42 x^3+173 x^2+61 x+141$
- $y^2=122 x^6+176 x^5+18 x^4+157 x^3+71 x^2+107 x+183$
- $y^2=188 x^6+63 x^5+47 x^4+97 x^3+30 x^2+82 x+172$
- $y^2=189 x^6+24 x^5+60 x^4+41 x^3+95 x^2+173 x+57$
- $y^2=125 x^6+184 x^5+97 x^4+127 x^3+15 x^2+62 x+190$
- $y^2=16 x^6+125 x^5+52 x^4+96 x^3+104 x^2+139 x+167$
- $y^2=85 x^6+43 x^5+68 x^4+192 x^3+196 x^2+20 x+142$
- $y^2=121 x^6+24 x^5+97 x^4+179 x^3+124 x^2+184 x+43$
- $y^2=138 x^6+190 x^5+157 x^4+73 x^3+172 x^2+54 x+149$
- $y^2=192 x^6+125 x^5+74 x^4+118 x^3+106 x^2+57 x+98$
- $y^2=172 x^6+68 x^5+146 x^4+179 x^3+163 x^2+107 x+101$
- $y^2=121 x^6+131 x^5+153 x^4+116 x^3+56 x^2+169 x+119$
- $y^2=68 x^6+60 x^5+110 x^4+15 x^3+123 x^2+176 x+36$
- $y^2=178 x^6+93 x^5+14 x^4+48 x^3+58 x^2+166 x+52$
- $y^2=123 x^6+20 x^5+19 x^4+166 x^3+106 x^2+37 x+113$
- $y^2=44 x^6+157 x^5+184 x^4+81 x^3+43 x^2+3 x+81$
- $y^2=158 x^6+173 x^5+49 x^4+123 x^3+114 x^2+87 x+32$
- $y^2=30 x^6+48 x^5+177 x^4+164 x^3+42 x^2+173 x+133$
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 4.0.15296400.2. |
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.bw_bkt | $2$ | (not in LMFDB) |