Properties

Label 14.189...921.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $1.897\times 10^{22}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(189\!\cdots\!921\)\(\medspace = 371131^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.371131.1
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 7.1.371131.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$: \( x^{2} + 221x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 a + 57 + \left(101 a + 4\right)\cdot 223 + 98\cdot 223^{2} + \left(91 a + 34\right)\cdot 223^{3} + \left(125 a + 89\right)\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 174 a + 59 + \left(86 a + 158\right)\cdot 223 + \left(116 a + 140\right)\cdot 223^{2} + \left(25 a + 36\right)\cdot 223^{3} + \left(33 a + 162\right)\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 83 + 56\cdot 223 + 204\cdot 223^{2} + 218\cdot 223^{3} + 116\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 49 a + 184 + \left(136 a + 157\right)\cdot 223 + \left(106 a + 63\right)\cdot 223^{2} + \left(197 a + 194\right)\cdot 223^{3} + \left(189 a + 202\right)\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 213 a + 77 + \left(121 a + 196\right)\cdot 223 + \left(222 a + 220\right)\cdot 223^{2} + \left(131 a + 215\right)\cdot 223^{3} + \left(97 a + 25\right)\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 155 + 129\cdot 223 + 145\cdot 223^{2} + 45\cdot 223^{3} + 136\cdot 223^{4} +O(223^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 + 189\cdot 223 + 18\cdot 223^{2} + 146\cdot 223^{3} + 158\cdot 223^{4} +O(223^{5})\) 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.