Invariants
| Base field: | $\F_{197}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 48 x + 959 x^{2} - 9456 x^{3} + 38809 x^{4}$ |
| Frobenius angles: | $\pm0.0739779530974$, $\pm0.236328988996$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.29150352.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $30265$ | $1491247345$ | $58445123686420$ | $2268498382872244425$ | $88036540833109044745825$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $150$ | $38424$ | $7644510$ | $1506168532$ | $296709764550$ | $58451729523486$ | $11514990388914750$ | $2268453121969730788$ | $446885265400029994950$ | $88036397287495071350664$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=78 x^6+163 x^5+77 x^4+189 x^3+2 x^2+93 x+102$
- $y^2=99 x^6+33 x^5+x^4+71 x^3+15 x^2+113 x+21$
- $y^2=x^6+69 x^5+42 x^4+152 x^3+16 x^2+10 x+158$
- $y^2=47 x^6+7 x^5+171 x^4+52 x^3+138 x^2+75 x+151$
- $y^2=127 x^6+99 x^5+119 x^4+51 x^3+48 x^2+96 x+125$
- $y^2=122 x^6+166 x^5+134 x^4+33 x^3+151 x^2+31 x+4$
- $y^2=66 x^6+21 x^5+65 x^4+83 x^3+41 x^2+185 x+132$
- $y^2=11 x^6+75 x^5+195 x^4+77 x^3+102 x^2+144 x+103$
- $y^2=29 x^6+156 x^5+95 x^4+68 x^3+2 x^2+127 x+165$
- $y^2=x^6+32 x^5+40 x^4+86 x^3+131 x^2+12 x+139$
- $y^2=102 x^6+3 x^5+56 x^4+146 x^3+104 x^2+22 x+154$
- $y^2=113 x^6+90 x^5+79 x^4+146 x^3+34 x^2+20 x+46$
- $y^2=150 x^6+77 x^5+83 x^4+180 x^3+140 x^2+x+156$
- $y^2=124 x^6+61 x^5+65 x^4+51 x^3+123 x^2+88 x+102$
- $y^2=100 x^6+177 x^5+169 x^4+105 x^3+61 x^2+43 x+87$
- $y^2=13 x^6+108 x^5+80 x^4+72 x^3+167 x^2+54 x+67$
- $y^2=99 x^6+7 x^5+56 x^4+27 x^3+190 x^2+148 x+184$
- $y^2=2 x^6+88 x^5+37 x^4+27 x^3+86 x^2+31 x+39$
- $y^2=167 x^6+142 x^5+125 x^4+126 x^3+12 x^2+24 x+10$
- $y^2=16 x^6+34 x^5+158 x^4+189 x^3+141 x^2+66 x+186$
- $y^2=79 x^6+184 x^5+184 x^4+119 x^3+174 x^2+26 x+105$
- $y^2=164 x^6+178 x^5+159 x^4+39 x^3+15 x^2+64 x+102$
- $y^2=120 x^6+190 x^5+190 x^4+64 x^3+5 x^2+41 x+150$
- $y^2=180 x^6+156 x^5+187 x^4+178 x^3+100 x^2+181 x+94$
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 4.0.29150352.1. |
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.bw_bkx | $2$ | (not in LMFDB) |