Properties

Label 14.144...871.30t565.a.a
Dimension $14$
Group $S_7$
Conductor $1.443\times 10^{49}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(144\!\cdots\!871\)\(\medspace = 109^{9} \cdot 2659^{9} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.289831.1
Galois orbit size: $1$
Smallest permutation container: 30T565
Parity: odd
Determinant: 1.289831.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.289831.1

Defining polynomial

$f(x)$$=$ \( x^{7} + x^{5} - x^{4} + x^{3} - x^{2} + x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: \( x^{2} + 82x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 81 a + 20 + \left(81 a + 80\right)\cdot 83 + \left(18 a + 17\right)\cdot 83^{2} + \left(44 a + 62\right)\cdot 83^{3} + \left(69 a + 17\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 40 + \left(41 a + 10\right)\cdot 83 + \left(43 a + 69\right)\cdot 83^{2} + \left(3 a + 66\right)\cdot 83^{3} + \left(14 a + 64\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 27 a + 30 + \left(21 a + 23\right)\cdot 83 + \left(2 a + 43\right)\cdot 83^{2} + \left(29 a + 8\right)\cdot 83^{3} + \left(70 a + 62\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 2 a + 18 + \left(a + 81\right)\cdot 83 + \left(64 a + 37\right)\cdot 83^{2} + \left(38 a + 4\right)\cdot 83^{3} + \left(13 a + 43\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 56 a + 57 + \left(61 a + 17\right)\cdot 83 + \left(80 a + 24\right)\cdot 83^{2} + \left(53 a + 35\right)\cdot 83^{3} + \left(12 a + 20\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 81 a + 42 + \left(41 a + 49\right)\cdot 83 + \left(39 a + 71\right)\cdot 83^{2} + \left(79 a + 26\right)\cdot 83^{3} + \left(68 a + 75\right)\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 42 + 69\cdot 83 + 67\cdot 83^{2} + 44\cdot 83^{3} + 48\cdot 83^{4} +O(83^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,2,3,4,5,6,7)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$14$
$21$$2$$(1,2)$$-4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$-1$
$280$$3$$(1,2,3)(4,5,6)$$2$
$210$$4$$(1,2,3,4)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$-1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$-1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$0$
$504$$10$$(1,2,3,4,5)(6,7)$$1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$

The blue line marks the conjugacy class containing complex conjugation.