Properties

Label 4.3_7e2_13e2_19e2.8t29.2c1
Dimension 4
Group $(((C_4 \times C_2): C_2):C_2):C_2$
Conductor $ 3 \cdot 7^{2} \cdot 13^{2} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:$8968323= 3 \cdot 7^{2} \cdot 13^{2} \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 41 x^{4} - 38 x^{3} - 64 x^{2} + 58 x + 53 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $(((C_4 \times C_2): C_2):C_2):C_2$
Parity: Odd
Determinant: 1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 523 }$ to precision 8.
Roots:
$r_{ 1 }$ $=$ $ 7 + 40\cdot 523 + 130\cdot 523^{2} + 522\cdot 523^{3} + 4\cdot 523^{4} + 470\cdot 523^{5} + 262\cdot 523^{6} + 389\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 33 + 374\cdot 523 + 20\cdot 523^{2} + 370\cdot 523^{3} + 420\cdot 523^{4} + 432\cdot 523^{5} + 448\cdot 523^{6} + 77\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 117 + 195\cdot 523 + 54\cdot 523^{2} + 358\cdot 523^{3} + 474\cdot 523^{4} + 55\cdot 523^{5} + 433\cdot 523^{6} + 286\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 124 + 149\cdot 523 + 153\cdot 523^{2} + 408\cdot 523^{3} + 280\cdot 523^{4} + 43\cdot 523^{5} + 44\cdot 523^{6} + 71\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 153 + 428\cdot 523 + 484\cdot 523^{2} + 415\cdot 523^{3} + 338\cdot 523^{4} + 367\cdot 523^{5} + 153\cdot 523^{6} + 304\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 375 + 469\cdot 523 + 447\cdot 523^{2} + 343\cdot 523^{3} + 348\cdot 523^{4} + 417\cdot 523^{5} + 427\cdot 523^{6} + 195\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 380 + 64\cdot 523 + 510\cdot 523^{2} + 253\cdot 523^{3} + 271\cdot 523^{4} + 12\cdot 523^{5} + 238\cdot 523^{6} + 141\cdot 523^{7} +O\left(523^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 381 + 370\cdot 523 + 290\cdot 523^{2} + 465\cdot 523^{3} + 474\cdot 523^{4} + 291\cdot 523^{5} + 83\cdot 523^{6} + 102\cdot 523^{7} +O\left(523^{ 8 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,8)(2,4)$
$(1,8)(2,5,4,7)$
$(1,5)(2,3)(4,6)(7,8)$
$(3,6)(5,7)$
$(2,4)(5,7)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,8)(2,4)(3,6)(5,7)$$-4$
$2$$2$$(1,6)(2,5)(3,8)(4,7)$$0$
$2$$2$$(1,8)(3,6)$$0$
$2$$2$$(1,6)(2,7)(3,8)(4,5)$$0$
$4$$2$$(1,5)(2,3)(4,6)(7,8)$$0$
$4$$2$$(3,6)(5,7)$$0$
$4$$2$$(1,3)(2,4)(5,7)(6,8)$$-2$
$4$$2$$(1,5)(2,6)(3,4)(7,8)$$0$
$4$$2$$(1,3)(6,8)$$2$
$4$$4$$(1,5,8,7)(2,3,4,6)$$0$
$4$$4$$(1,5,8,7)(2,6,4,3)$$0$
$4$$4$$(1,6,8,3)(2,7,4,5)$$0$
$8$$4$$(1,4,6,7)(2,3,5,8)$$0$
$8$$4$$(1,6,8,3)(5,7)$$0$
$8$$4$$(1,2,6,7)(3,5,8,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.