Properties

Label 4.37e2_59e2.8t23.1c1
Dimension 4
Group $\textrm{GL(2,3)}$
Conductor $ 37^{2} \cdot 59^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$\textrm{GL(2,3)}$
Conductor:$4765489= 37^{2} \cdot 59^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 8 x^{6} + 15 x^{5} + 20 x^{4} - 46 x^{3} - 17 x^{2} + 74 x - 37 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\textrm{GL(2,3)}$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 6 + 33\cdot 79 + 23\cdot 79^{2} + 61\cdot 79^{3} + 9\cdot 79^{4} + 27\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 6 a + 77 + \left(36 a + 1\right)\cdot 79 + \left(23 a + 17\right)\cdot 79^{2} + \left(64 a + 39\right)\cdot 79^{3} + \left(75 a + 75\right)\cdot 79^{4} + \left(25 a + 56\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 35 a + 17 + \left(18 a + 17\right)\cdot 79 + \left(6 a + 41\right)\cdot 79^{2} + \left(17 a + 51\right)\cdot 79^{3} + \left(42 a + 74\right)\cdot 79^{4} + \left(42 a + 59\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 51 a + 27 + \left(68 a + 32\right)\cdot 79 + \left(33 a + 59\right)\cdot 79^{2} + \left(54 a + 64\right)\cdot 79^{3} + \left(66 a + 51\right)\cdot 79^{4} + \left(75 a + 59\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 73 a + 4 + \left(42 a + 32\right)\cdot 79 + \left(55 a + 4\right)\cdot 79^{2} + \left(14 a + 1\right)\cdot 79^{3} + \left(3 a + 8\right)\cdot 79^{4} + \left(53 a + 7\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 44 a + 52 + 60 a\cdot 79 + \left(72 a + 29\right)\cdot 79^{2} + \left(61 a + 62\right)\cdot 79^{3} + \left(36 a + 20\right)\cdot 79^{4} + \left(36 a + 60\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 56 + 69\cdot 79 + 37\cdot 79^{2} + 29\cdot 79^{3} + 11\cdot 79^{4} + 55\cdot 79^{5} +O\left(79^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 28 a + 78 + \left(10 a + 49\right)\cdot 79 + \left(45 a + 24\right)\cdot 79^{2} + \left(24 a + 6\right)\cdot 79^{3} + \left(12 a + 64\right)\cdot 79^{4} + \left(3 a + 68\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,7)(2,3)(4,8)(5,6)$$-4$
$12$$2$$(2,3)(4,5)(6,8)$$0$
$8$$3$$(1,6,8)(4,7,5)$$1$
$6$$4$$(1,4,7,8)(2,6,3,5)$$0$
$8$$6$$(1,7)(2,8,6,3,4,5)$$-1$
$6$$8$$(1,5,3,4,7,6,2,8)$$0$
$6$$8$$(1,6,3,8,7,5,2,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.