Properties

Label 8.2e10_3e12_13e4.36t555.2c2
Dimension 8
Group $A_6$
Conductor $ 2^{10} \cdot 3^{12} \cdot 13^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$8$
Group:$A_6$
Conductor:$15542770074624= 2^{10} \cdot 3^{12} \cdot 13^{4} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 2 $ over $\Q$
Size of Galois orbit: 2
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 $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 61 a + 28 + \left(14 a + 25\right)\cdot 73 + \left(57 a + 30\right)\cdot 73^{2} + \left(34 a + 65\right)\cdot 73^{3} + \left(15 a + 71\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 60 + 46\cdot 73 + 62\cdot 73^{2} + 17\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 65 + \left(58 a + 8\right)\cdot 73 + \left(15 a + 41\right)\cdot 73^{2} + \left(38 a + 39\right)\cdot 73^{3} + \left(57 a + 10\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 61 + 21\cdot 73 + 68\cdot 73^{2} + 67\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 72 a + 42 + \left(41 a + 67\right)\cdot 73 + \left(27 a + 60\right)\cdot 73^{2} + \left(44 a + 70\right)\cdot 73^{3} + \left(63 a + 20\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ a + 39 + \left(31 a + 48\right)\cdot 73 + \left(45 a + 28\right)\cdot 73^{2} + \left(28 a + 30\right)\cdot 73^{3} + \left(9 a + 21\right)\cdot 73^{4} +O\left(73^{ 5 }\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$$1$$()$$8$
$45$$2$$(1,2)(3,4)$$0$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$72$$5$$(1,2,3,4,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
$72$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.