Properties

Label 4.3e2_7e3_13e3_19e3.8t35.2c1
Dimension 4
Group $C_2 \wr C_2\wr C_2$
Conductor $ 3^{2} \cdot 7^{3} \cdot 13^{3} \cdot 19^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$C_2 \wr C_2\wr C_2$
Conductor:$46518691401= 3^{2} \cdot 7^{3} \cdot 13^{3} \cdot 19^{3} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + x^{5} - x^{4} - 2 x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2 \wr C_2\wr C_2$
Parity: Even
Determinant: 1.7_13_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 14 a + 7 + \left(18 a + 64\right)\cdot 67 + \left(38 a + 1\right)\cdot 67^{2} + \left(55 a + 40\right)\cdot 67^{3} + \left(60 a + 41\right)\cdot 67^{4} + 58\cdot 67^{5} + \left(13 a + 46\right)\cdot 67^{6} + \left(26 a + 62\right)\cdot 67^{7} + \left(24 a + 49\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 32 + 3\cdot 67 + 48\cdot 67^{2} + 59\cdot 67^{3} + 10\cdot 67^{4} + 12\cdot 67^{5} + 46\cdot 67^{6} + 5\cdot 67^{7} + 16\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 40 a + 6 + \left(59 a + 51\right)\cdot 67 + \left(36 a + 20\right)\cdot 67^{2} + \left(29 a + 32\right)\cdot 67^{3} + \left(15 a + 54\right)\cdot 67^{4} + \left(52 a + 27\right)\cdot 67^{5} + \left(33 a + 17\right)\cdot 67^{6} + \left(49 a + 23\right)\cdot 67^{7} + \left(44 a + 63\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 27 a + 32 + \left(7 a + 48\right)\cdot 67 + \left(30 a + 41\right)\cdot 67^{2} + \left(37 a + 46\right)\cdot 67^{3} + \left(51 a + 19\right)\cdot 67^{4} + \left(14 a + 20\right)\cdot 67^{5} + \left(33 a + 33\right)\cdot 67^{6} + \left(17 a + 53\right)\cdot 67^{7} + \left(22 a + 58\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 3 + 59\cdot 67^{2} + 11\cdot 67^{3} + 11\cdot 67^{4} + 23\cdot 67^{5} + 54\cdot 67^{6} + 26\cdot 67^{7} + 7\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 53 a + 63 + \left(48 a + 55\right)\cdot 67 + \left(28 a + 2\right)\cdot 67^{2} + \left(11 a + 23\right)\cdot 67^{3} + \left(6 a + 28\right)\cdot 67^{4} + \left(66 a + 1\right)\cdot 67^{5} + \left(53 a + 31\right)\cdot 67^{6} + \left(40 a + 20\right)\cdot 67^{7} + \left(42 a + 54\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 7 }$ $=$ $ 48 a + 34 + \left(36 a + 6\right)\cdot 67 + \left(28 a + 8\right)\cdot 67^{2} + \left(28 a + 18\right)\cdot 67^{3} + \left(13 a + 38\right)\cdot 67^{4} + \left(65 a + 5\right)\cdot 67^{5} + \left(16 a + 18\right)\cdot 67^{6} + \left(43 a + 60\right)\cdot 67^{7} + \left(60 a + 9\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$
$r_{ 8 }$ $=$ $ 19 a + 25 + \left(30 a + 38\right)\cdot 67 + \left(38 a + 18\right)\cdot 67^{2} + \left(38 a + 36\right)\cdot 67^{3} + \left(53 a + 63\right)\cdot 67^{4} + \left(a + 51\right)\cdot 67^{5} + \left(50 a + 20\right)\cdot 67^{6} + \left(23 a + 15\right)\cdot 67^{7} + \left(6 a + 8\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,6)(2,5)(3,8)(4,7)$$-4$
$2$$2$$(3,8)(4,7)$$0$
$4$$2$$(2,5)(4,7)$$0$
$4$$2$$(3,8)$$-2$
$4$$2$$(1,2)(3,7)(4,8)(5,6)$$0$
$4$$2$$(1,6)(2,5)(3,7)(4,8)$$2$
$4$$2$$(3,7)(4,8)$$-2$
$4$$2$$(1,6)(2,5)(3,8)$$2$
$8$$2$$(2,5)(3,7)(4,8)$$0$
$8$$2$$(1,4)(2,3)(5,8)(6,7)$$0$
$4$$4$$(1,5,6,2)(3,4,8,7)$$0$
$4$$4$$(3,4,8,7)$$2$
$4$$4$$(1,5,6,2)(3,8)(4,7)$$-2$
$8$$4$$(1,3,6,8)(2,4,5,7)$$0$
$8$$4$$(1,6)(3,4,8,7)$$0$
$8$$4$$(1,5,6,2)(3,4)(7,8)$$0$
$16$$4$$(1,3)(2,4,5,7)(6,8)$$0$
$16$$4$$(1,3,2,7)(4,6,8,5)$$0$
$16$$8$$(1,3,5,4,6,8,2,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.