Properties

Label 14.392...721.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $3.929\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(392\!\cdots\!721\)\(\medspace = 13^{4} \cdot 19259^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.250367.1
Galois orbit size: $1$
Smallest permutation container: 21T38
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.250367.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \( x^{2} + 97x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 18 a + 14 + \left(24 a + 17\right)\cdot 101 + \left(82 a + 3\right)\cdot 101^{2} + \left(12 a + 63\right)\cdot 101^{3} + \left(69 a + 59\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 74 a + 31 + \left(54 a + 26\right)\cdot 101 + \left(70 a + 80\right)\cdot 101^{2} + \left(16 a + 100\right)\cdot 101^{3} + 24\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 87 + \left(43 a + 68\right)\cdot 101 + \left(17 a + 44\right)\cdot 101^{2} + \left(9 a + 95\right)\cdot 101^{3} + \left(34 a + 91\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 83 a + 86 + \left(76 a + 95\right)\cdot 101 + \left(18 a + 4\right)\cdot 101^{2} + \left(88 a + 32\right)\cdot 101^{3} + \left(31 a + 20\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 69 + 88\cdot 101 + 93\cdot 101^{2} + 101^{3} + 80\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 24 + \left(46 a + 70\right)\cdot 101 + \left(30 a + 4\right)\cdot 101^{2} + \left(84 a + 97\right)\cdot 101^{3} + \left(100 a + 8\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 99 a + 95 + \left(57 a + 36\right)\cdot 101 + \left(83 a + 71\right)\cdot 101^{2} + \left(91 a + 13\right)\cdot 101^{3} + \left(66 a + 17\right)\cdot 101^{4} +O(101^{5})\) 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$$(1,2)$$6$
$105$$2$$(1,2)(3,4)(5,6)$$2$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$-1$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$-1$
$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)$$0$

The blue line marks the conjugacy class containing complex conjugation.