Properties

Label 4.5e2_569.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 569 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$14225= 5^{2} \cdot 569 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} + 3 x^{3} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.569.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 37 a + 9 + \left(56 a + 19\right)\cdot 61 + \left(58 a + 10\right)\cdot 61^{2} + \left(53 a + 23\right)\cdot 61^{3} + \left(57 a + 48\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 27 + 54\cdot 61 + 32\cdot 61^{2} + 22\cdot 61^{3} + 55\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 12 + \left(21 a + 59\right)\cdot 61 + \left(19 a + 14\right)\cdot 61^{2} + \left(58 a + 30\right)\cdot 61^{3} + \left(25 a + 49\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 a + 46 + \left(4 a + 38\right)\cdot 61 + \left(2 a + 12\right)\cdot 61^{2} + \left(7 a + 18\right)\cdot 61^{3} + \left(3 a + 52\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 + 3\cdot 61 + 38\cdot 61^{2} + 19\cdot 61^{3} + 21\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 50 a + 23 + \left(39 a + 8\right)\cdot 61 + \left(41 a + 13\right)\cdot 61^{2} + \left(2 a + 8\right)\cdot 61^{3} + \left(35 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,4)(5,6)$$0$
$6$$2$$(3,6)$$2$
$9$$2$$(3,6)(4,5)$$0$
$4$$3$$(1,4,5)$$1$
$4$$3$$(1,4,5)(2,3,6)$$-2$
$18$$4$$(1,2)(3,5,6,4)$$0$
$12$$6$$(1,3,4,6,5,2)$$0$
$12$$6$$(1,4,5)(3,6)$$-1$
The blue line marks the conjugacy class containing complex conjugation.