Properties

Label 4.5_11e2_29e2.8t29.3c1
Dimension 4
Group $(((C_4 \times C_2): C_2):C_2):C_2$
Conductor $ 5 \cdot 11^{2} \cdot 29^{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:$508805= 5 \cdot 11^{2} \cdot 29^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $(((C_4 \times C_2): C_2):C_2):C_2$
Parity: Even
Determinant: 1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ $ 31 + 345\cdot 929 + 280\cdot 929^{2} + 51\cdot 929^{3} + 355\cdot 929^{4} + 762\cdot 929^{5} + 249\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 140 + 502\cdot 929 + 7\cdot 929^{2} + 62\cdot 929^{3} + 107\cdot 929^{4} + 18\cdot 929^{5} + 279\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 151 + 749\cdot 929 + 698\cdot 929^{2} + 11\cdot 929^{3} + 466\cdot 929^{4} + 220\cdot 929^{5} + 41\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 591 + 857\cdot 929 + 639\cdot 929^{2} + 212\cdot 929^{3} + 823\cdot 929^{4} + 223\cdot 929^{5} + 250\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 595 + 595\cdot 929 + 818\cdot 929^{2} + 210\cdot 929^{3} + 452\cdot 929^{4} + 756\cdot 929^{5} + 702\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 676 + 630\cdot 929 + 522\cdot 929^{2} + 62\cdot 929^{3} + 287\cdot 929^{4} + 283\cdot 929^{5} + 254\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 740 + 158\cdot 929 + 686\cdot 929^{2} + 102\cdot 929^{3} + 767\cdot 929^{4} + 370\cdot 929^{5} + 441\cdot 929^{6} +O\left(929^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 793 + 805\cdot 929 + 61\cdot 929^{2} + 215\cdot 929^{3} + 458\cdot 929^{4} + 151\cdot 929^{5} + 568\cdot 929^{6} +O\left(929^{ 7 }\right)$

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

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

Character values on conjugacy classes

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