Properties

Label 14.165...081.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $1.654\times 10^{21}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(165\!\cdots\!081\)\(\medspace = 17^{4} \cdot 11863^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.201671.1
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 7.1.201671.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: \( x^{2} + 274x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 168 a + 90 + \left(208 a + 2\right)\cdot 277 + \left(236 a + 96\right)\cdot 277^{2} + \left(152 a + 64\right)\cdot 277^{3} + \left(129 a + 87\right)\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 177 a + 107 + \left(86 a + 169\right)\cdot 277 + \left(32 a + 270\right)\cdot 277^{2} + \left(165 a + 183\right)\cdot 277^{3} + \left(63 a + 168\right)\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 167 a + 114 + \left(265 a + 45\right)\cdot 277 + \left(177 a + 249\right)\cdot 277^{2} + \left(125 a + 27\right)\cdot 277^{3} + \left(57 a + 75\right)\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 100 a + 84 + \left(190 a + 252\right)\cdot 277 + \left(244 a + 3\right)\cdot 277^{2} + \left(111 a + 93\right)\cdot 277^{3} + \left(213 a + 194\right)\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 59 + 57\cdot 277 + 204\cdot 277^{2} + 225\cdot 277^{3} + 137\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 109 a + 40 + \left(68 a + 183\right)\cdot 277 + \left(40 a + 43\right)\cdot 277^{2} + \left(124 a + 9\right)\cdot 277^{3} + \left(147 a + 46\right)\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 110 a + 61 + \left(11 a + 121\right)\cdot 277 + \left(99 a + 240\right)\cdot 277^{2} + \left(151 a + 226\right)\cdot 277^{3} + \left(219 a + 121\right)\cdot 277^{4} +O(277^{5})\) 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.