Properties

Label 4.5e2_281.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 281 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$7025= 5^{2} \cdot 281 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - x^{4} + 2 x^{3} - x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.281.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 41 a + 27 + \left(44 a + 14\right)\cdot 59 + \left(32 a + 5\right)\cdot 59^{2} + \left(27 a + 9\right)\cdot 59^{3} + \left(34 a + 36\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 a + 9 + \left(14 a + 18\right)\cdot 59 + \left(26 a + 52\right)\cdot 59^{2} + \left(31 a + 3\right)\cdot 59^{3} + \left(24 a + 43\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 49 + 20\cdot 59 + 27\cdot 59^{2} + 32\cdot 59^{3} + 16\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 8 a + 57 + 2\cdot 59 + \left(56 a + 57\right)\cdot 59^{2} + \left(a + 32\right)\cdot 59^{3} + \left(13 a + 36\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 51 a + 6 + \left(58 a + 54\right)\cdot 59 + \left(2 a + 53\right)\cdot 59^{2} + \left(57 a + 37\right)\cdot 59^{3} + \left(45 a + 47\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 30 + 7\cdot 59 + 40\cdot 59^{2} + 59^{3} + 56\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

Character values on conjugacy classes

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