Properties

Label 1.91.6t1.h.b
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 6.6.891474493.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{91}(4,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 31x^{4} + 4x^{3} + 253x^{2} + 101x - 391 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 13 a + 33 + \left(3 a + 27\right)\cdot 37 + \left(20 a + 34\right)\cdot 37^{2} + \left(5 a + 6\right)\cdot 37^{3} + \left(17 a + 9\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a + 11 + \left(33 a + 28\right)\cdot 37 + 16 a\cdot 37^{2} + \left(31 a + 9\right)\cdot 37^{3} + \left(19 a + 35\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 a + 10 + \left(18 a + 34\right)\cdot 37 + 18\cdot 37^{2} + \left(35 a + 10\right)\cdot 37^{3} + \left(24 a + 9\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 3 + \left(36 a + 9\right)\cdot 37 + \left(14 a + 15\right)\cdot 37^{2} + \left(30 a + 6\right)\cdot 37^{3} + \left(8 a + 26\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 26 a + 10 + 32\cdot 37 + \left(22 a + 1\right)\cdot 37^{2} + \left(6 a + 2\right)\cdot 37^{3} + \left(28 a + 31\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 19 a + 8 + \left(18 a + 16\right)\cdot 37 + \left(36 a + 2\right)\cdot 37^{2} + \left(a + 2\right)\cdot 37^{3} + 12 a\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.