Properties

Label 14.201...721.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $2.014\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(201\!\cdots\!721\)\(\medspace = 19^{4} \cdot 11149^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.211831.1
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 7.1.211831.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: \( x^{2} + 261x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 261 + 63\cdot 263 + 35\cdot 263^{2} + 254\cdot 263^{3} + 54\cdot 263^{4} +O(263^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 228 a + 251 + \left(258 a + 231\right)\cdot 263 + \left(61 a + 97\right)\cdot 263^{2} + \left(184 a + 242\right)\cdot 263^{3} + \left(15 a + 215\right)\cdot 263^{4} +O(263^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 85 a + 177 + \left(48 a + 161\right)\cdot 263 + \left(5 a + 34\right)\cdot 263^{2} + \left(226 a + 27\right)\cdot 263^{3} + \left(260 a + 223\right)\cdot 263^{4} +O(263^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 184 + 198\cdot 263 + 69\cdot 263^{2} + 140\cdot 263^{3} +O(263^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 35 a + 181 + \left(4 a + 258\right)\cdot 263 + \left(201 a + 225\right)\cdot 263^{2} + \left(78 a + 22\right)\cdot 263^{3} + \left(247 a + 63\right)\cdot 263^{4} +O(263^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 178 a + 84 + \left(214 a + 173\right)\cdot 263 + \left(257 a + 259\right)\cdot 263^{2} + \left(36 a + 210\right)\cdot 263^{3} + \left(2 a + 255\right)\cdot 263^{4} +O(263^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 178 + 226\cdot 263 + 65\cdot 263^{2} + 154\cdot 263^{3} + 238\cdot 263^{4} +O(263^{5})\) 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.