Properties

Label 14.470...881.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $4.703\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(470\!\cdots\!881\)\(\medspace = 307^{4} \cdot 853^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.261871.1
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 7.1.261871.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 + 31\cdot 41 + 35\cdot 41^{2} + 10\cdot 41^{3} + 35\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 11 + \left(8 a + 21\right)\cdot 41 + \left(12 a + 11\right)\cdot 41^{2} + \left(21 a + 15\right)\cdot 41^{3} + \left(2 a + 8\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 34 a + 32 + \left(32 a + 38\right)\cdot 41 + \left(28 a + 39\right)\cdot 41^{2} + \left(19 a + 25\right)\cdot 41^{3} + \left(38 a + 35\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 a + \left(29 a + 24\right)\cdot 41 + \left(8 a + 24\right)\cdot 41^{2} + \left(a + 10\right)\cdot 41^{3} + \left(39 a + 24\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 25 + \left(11 a + 8\right)\cdot 41 + \left(32 a + 21\right)\cdot 41^{2} + \left(39 a + 5\right)\cdot 41^{3} + \left(a + 17\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 37 + 19\cdot 41 + 39\cdot 41^{2} + 16\cdot 41^{3} + 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 8 + 20\cdot 41 + 32\cdot 41^{2} + 37\cdot 41^{3} +O(41^{5})\) 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.