Basic invariants
| Dimension: | $70$ |
| Group: | $A_8$ |
| Conductor: | \(590\!\cdots\!104\)\(\medspace = 2^{222} \cdot 43^{60} \cdot 107^{60} \cdot 179^{60} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.81803428774472904272307991671908472193024.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 120 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.81803428774472904272307991671908472193024.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$:
\( x^{2} + 306x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 303 a + 119 + \left(115 a + 212\right)\cdot 307 + \left(238 a + 19\right)\cdot 307^{2} + \left(217 a + 39\right)\cdot 307^{3} + \left(104 a + 185\right)\cdot 307^{4} + \left(56 a + 247\right)\cdot 307^{5} + \left(238 a + 213\right)\cdot 307^{6} + \left(238 a + 73\right)\cdot 307^{7} + \left(196 a + 295\right)\cdot 307^{8} + \left(4 a + 106\right)\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 281 a + 246 + \left(73 a + 233\right)\cdot 307 + \left(135 a + 47\right)\cdot 307^{2} + \left(43 a + 134\right)\cdot 307^{3} + \left(228 a + 141\right)\cdot 307^{4} + \left(250 a + 230\right)\cdot 307^{5} + \left(30 a + 115\right)\cdot 307^{6} + \left(262 a + 159\right)\cdot 307^{7} + \left(205 a + 62\right)\cdot 307^{8} + \left(114 a + 46\right)\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 152 + 240\cdot 307 + 211\cdot 307^{2} + 270\cdot 307^{3} + 79\cdot 307^{4} + 60\cdot 307^{5} + 39\cdot 307^{6} + 41\cdot 307^{7} + 24\cdot 307^{8} + 10\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 115 + \left(191 a + 25\right)\cdot 307 + \left(68 a + 142\right)\cdot 307^{2} + \left(89 a + 18\right)\cdot 307^{3} + \left(202 a + 72\right)\cdot 307^{4} + \left(250 a + 199\right)\cdot 307^{5} + \left(68 a + 88\right)\cdot 307^{6} + \left(68 a + 74\right)\cdot 307^{7} + \left(110 a + 253\right)\cdot 307^{8} + \left(302 a + 221\right)\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 150 + 68\cdot 307 + 274\cdot 307^{2} + 301\cdot 307^{3} + 212\cdot 307^{4} + 16\cdot 307^{5} + 48\cdot 307^{6} + 196\cdot 307^{7} + 2\cdot 307^{8} + 223\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 65 + 304\cdot 307 + 293\cdot 307^{2} + 301\cdot 307^{3} + 246\cdot 307^{4} + 51\cdot 307^{5} + 187\cdot 307^{6} + 144\cdot 307^{7} + 50\cdot 307^{8} + 60\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 a + 220 + \left(233 a + 26\right)\cdot 307 + \left(171 a + 109\right)\cdot 307^{2} + \left(263 a + 42\right)\cdot 307^{3} + \left(78 a + 19\right)\cdot 307^{4} + \left(56 a + 253\right)\cdot 307^{5} + \left(276 a + 202\right)\cdot 307^{6} + \left(44 a + 83\right)\cdot 307^{7} + \left(101 a + 6\right)\cdot 307^{8} + \left(192 a + 262\right)\cdot 307^{9} +O(307^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 161 + 116\cdot 307 + 129\cdot 307^{2} + 119\cdot 307^{3} + 270\cdot 307^{4} + 168\cdot 307^{5} + 25\cdot 307^{6} + 148\cdot 307^{7} + 226\cdot 307^{8} + 297\cdot 307^{9} +O(307^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $70$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $112$ | $3$ | $(1,2,3)$ | $-5$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $-2$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |