Properties

Label 20.150...601.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.509\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(150\!\cdots\!601\)\(\medspace = 11^{10} \cdot 41^{10} \cdot 461^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.207911.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.207911.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: \( x^{2} + 152x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 44 a + 152 + \left(119 a + 131\right)\cdot 157 + \left(34 a + 89\right)\cdot 157^{2} + \left(123 a + 101\right)\cdot 157^{3} + \left(112 a + 63\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 53 a + 74 + \left(62 a + 109\right)\cdot 157 + \left(5 a + 122\right)\cdot 157^{2} + \left(80 a + 49\right)\cdot 157^{3} + \left(81 a + 67\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 134 a + 31 + \left(82 a + 25\right)\cdot 157 + \left(152 a + 20\right)\cdot 157^{2} + \left(124 a + 156\right)\cdot 157^{3} + \left(133 a + 125\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 104 a + 25 + \left(94 a + 54\right)\cdot 157 + \left(151 a + 87\right)\cdot 157^{2} + \left(76 a + 130\right)\cdot 157^{3} + \left(75 a + 80\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 58 + 102\cdot 157 + 91\cdot 157^{2} + 134\cdot 157^{3} + 57\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 23 a + 73 + \left(74 a + 148\right)\cdot 157 + \left(4 a + 71\right)\cdot 157^{2} + 32 a\cdot 157^{3} + \left(23 a + 42\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 113 a + 58 + \left(37 a + 56\right)\cdot 157 + \left(122 a + 144\right)\cdot 157^{2} + \left(33 a + 54\right)\cdot 157^{3} + \left(44 a + 33\right)\cdot 157^{4} +O(157^{5})\) 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$$1$$()$$20$
$21$$2$$(1,2)$$0$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$-4$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$2$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.