Properties

Label 5.3e10_5e8.6t15.2c1
Dimension 5
Group $A_6$
Conductor $ 3^{10} \cdot 5^{8}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$23066015625= 3^{10} \cdot 5^{8} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 15 x^{4} - 25 x^{3} - 45 x + 60 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 75 a + 87 + \left(5 a + 44\right)\cdot 89 + \left(56 a + 10\right)\cdot 89^{2} + \left(53 a + 31\right)\cdot 89^{3} + \left(75 a + 1\right)\cdot 89^{4} + \left(52 a + 13\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 45 + \left(22 a + 45\right)\cdot 89 + \left(20 a + 25\right)\cdot 89^{2} + \left(72 a + 17\right)\cdot 89^{3} + \left(69 a + 77\right)\cdot 89^{4} + \left(31 a + 83\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 86 a + 66 + \left(66 a + 18\right)\cdot 89 + \left(68 a + 56\right)\cdot 89^{2} + \left(16 a + 57\right)\cdot 89^{3} + \left(19 a + 48\right)\cdot 89^{4} + \left(57 a + 58\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 68 + 87\cdot 89 + 21\cdot 89^{2} + 12\cdot 89^{3} + 4\cdot 89^{4} + 38\cdot 89^{5} +O\left(89^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 14 a + 78 + \left(83 a + 10\right)\cdot 89 + \left(32 a + 41\right)\cdot 89^{2} + \left(35 a + 83\right)\cdot 89^{3} + \left(13 a + 31\right)\cdot 89^{4} + \left(36 a + 40\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 15 + 59\cdot 89 + 22\cdot 89^{2} + 65\cdot 89^{3} + 14\cdot 89^{4} + 33\cdot 89^{5} +O\left(89^{ 6 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$72$$5$$(1,2,3,4,5)$$0$
$72$$5$$(1,3,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.