Properties

Label 14.832...507.30t565.a.a
Dimension $14$
Group $S_7$
Conductor $8.325\times 10^{26}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(832\!\cdots\!507\)\(\medspace = 242147^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.242147.1
Galois orbit size: $1$
Smallest permutation container: 30T565
Parity: odd
Determinant: 1.242147.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.242147.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 2x^{5} - x^{4} - x^{3} + 2x^{2} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{2} + 82x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 67 a + 27 + \left(5 a + 4\right)\cdot 89 + \left(52 a + 6\right)\cdot 89^{2} + \left(38 a + 84\right)\cdot 89^{3} + \left(32 a + 60\right)\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 82 a + \left(41 a + 25\right)\cdot 89 + \left(23 a + 55\right)\cdot 89^{2} + \left(26 a + 58\right)\cdot 89^{3} + \left(23 a + 22\right)\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a + 40 + \left(47 a + 58\right)\cdot 89 + \left(65 a + 88\right)\cdot 89^{2} + \left(62 a + 40\right)\cdot 89^{3} + \left(65 a + 70\right)\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a + 51 + \left(83 a + 66\right)\cdot 89 + \left(36 a + 8\right)\cdot 89^{2} + \left(50 a + 35\right)\cdot 89^{3} + \left(56 a + 71\right)\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O(89^{5})\) 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.