Properties

Label 5.13e2_19e4.6t15.2c1
Dimension 5
Group $A_6$
Conductor $ 13^{2} \cdot 19^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$22024249= 13^{2} \cdot 19^{4} $
Artin number field: Splitting field of $f= x^{6} - 4 x^{4} - 15 x^{3} - 15 x^{2} - 8 x + 4 $ 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 $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 6 a + 33 + \left(70 a + 4\right)\cdot 73 + \left(62 a + 45\right)\cdot 73^{2} + \left(59 a + 29\right)\cdot 73^{3} + \left(25 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 24 + 44\cdot 73 + 38\cdot 73^{2} + 28\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 48 + 50\cdot 73 + 26\cdot 73^{2} + 22\cdot 73^{3} + 41\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 36 a + 14 + \left(58 a + 68\right)\cdot 73 + \left(71 a + 39\right)\cdot 73^{2} + \left(36 a + 49\right)\cdot 73^{3} + \left(22 a + 53\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 37 a + 49 + \left(14 a + 61\right)\cdot 73 + \left(a + 50\right)\cdot 73^{2} + \left(36 a + 15\right)\cdot 73^{3} + \left(50 a + 11\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 67 a + 51 + \left(2 a + 62\right)\cdot 73 + \left(10 a + 17\right)\cdot 73^{2} + 13 a\cdot 73^{3} + \left(47 a + 54\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$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$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.