Properties

Label 1.91.6t1.d.a
Dimension $1$
Group $C_6$
Conductor $91$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(91\)\(\medspace = 7 \cdot 13 \)
Artin field: Galois closure of 6.0.36924979.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{91}(38,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 22x^{4} - 22x^{3} + 148x^{2} - 148x + 337 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{2} + 69x + 7 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 63 a + 53 + \left(39 a + 56\right)\cdot 71 + \left(12 a + 68\right)\cdot 71^{2} + \left(35 a + 49\right)\cdot 71^{3} + \left(31 a + 44\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 61 a + 8 + \left(a + 41\right)\cdot 71 + \left(54 a + 61\right)\cdot 71^{2} + \left(57 a + 44\right)\cdot 71^{3} + \left(65 a + 51\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 59 + \left(69 a + 54\right)\cdot 71 + \left(16 a + 25\right)\cdot 71^{2} + \left(13 a + 35\right)\cdot 71^{3} + \left(5 a + 54\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 40 a + 24 + \left(26 a + 22\right)\cdot 71 + \left(25 a + 60\right)\cdot 71^{2} + \left(65 a + 5\right)\cdot 71^{3} + \left(7 a + 55\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 37 + \left(31 a + 2\right)\cdot 71 + \left(58 a + 54\right)\cdot 71^{2} + \left(35 a + 36\right)\cdot 71^{3} + \left(39 a + 1\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 31 a + 33 + \left(44 a + 35\right)\cdot 71 + \left(45 a + 13\right)\cdot 71^{2} + \left(5 a + 40\right)\cdot 71^{3} + \left(63 a + 5\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,5)(2,3)(4,6)$$-1$
$1$$3$$(1,3,4)(2,6,5)$$\zeta_{3}$
$1$$3$$(1,4,3)(2,5,6)$$-\zeta_{3} - 1$
$1$$6$$(1,6,3,5,4,2)$$\zeta_{3} + 1$
$1$$6$$(1,2,4,5,3,6)$$-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.