Properties

Label 14.936...681.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $9.363\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(936\!\cdots\!681\)\(\medspace = 277^{4} \cdot 1123^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.311071.1
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 7.1.311071.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{2} + 96x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 + 75\cdot 97 + 45\cdot 97^{2} + 83\cdot 97^{3} + 21\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 54 a + 47 + \left(16 a + 94\right)\cdot 97 + \left(9 a + 77\right)\cdot 97^{2} + \left(54 a + 62\right)\cdot 97^{3} + \left(20 a + 93\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 a + 4 + \left(80 a + 57\right)\cdot 97 + \left(87 a + 70\right)\cdot 97^{2} + \left(42 a + 10\right)\cdot 97^{3} + \left(76 a + 60\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 87 a + 25 + \left(64 a + 27\right)\cdot 97 + \left(4 a + 58\right)\cdot 97^{2} + \left(29 a + 58\right)\cdot 97^{3} + \left(53 a + 63\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 15 + \left(32 a + 5\right)\cdot 97 + \left(92 a + 95\right)\cdot 97^{2} + \left(67 a + 82\right)\cdot 97^{3} + \left(43 a + 87\right)\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 41 + 97 + 15\cdot 97^{2} + 97^{3} + 41\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 55 + 30\cdot 97 + 25\cdot 97^{2} + 88\cdot 97^{3} + 19\cdot 97^{4} +O(97^{5})\) 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.