Properties

Label 10.11e4_41e6.30t176.2c1
Dimension 10
Group $S_6$
Conductor $ 11^{4} \cdot 41^{6}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$69546276192481= 11^{4} \cdot 41^{6} $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 4 x^{4} + 6 x^{3} - 6 x + 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 30 a + 38 + 14\cdot 53 + \left(48 a + 3\right)\cdot 53^{2} + \left(40 a + 43\right)\cdot 53^{3} + 4 a\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 a + 52 + \left(52 a + 39\right)\cdot 53 + \left(4 a + 35\right)\cdot 53^{2} + \left(12 a + 52\right)\cdot 53^{3} + \left(48 a + 31\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 52\cdot 53 + 30\cdot 53^{2} + 48\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 51 a + 40 + \left(12 a + 26\right)\cdot 53 + \left(33 a + 39\right)\cdot 53^{2} + \left(6 a + 40\right)\cdot 53^{3} + \left(52 a + 52\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 2 a + 32 + \left(40 a + 27\right)\cdot 53 + 19 a\cdot 53^{2} + \left(46 a + 34\right)\cdot 53^{3} + 42\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 27 + 50\cdot 53 + 48\cdot 53^{2} + 45\cdot 53^{3} + 32\cdot 53^{4} +O\left(53^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$2$
$15$$2$$(1,2)$$-2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$1$
The blue line marks the conjugacy class containing complex conjugation.