Properties

Label 20.691...849.70.a.a
Dimension $20$
Group $S_7$
Conductor $6.915\times 10^{55}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(691\!\cdots\!849\)\(\medspace = 31^{10} \cdot 12377^{10} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.383687.1
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 7.1.383687.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{2} + 70x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 + 60\cdot 73 + 52\cdot 73^{2} + 34\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 34 + 33\cdot 73 + 71\cdot 73^{2} + 7\cdot 73^{3} + 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 38 a + 66 + \left(63 a + 43\right)\cdot 73 + \left(49 a + 24\right)\cdot 73^{2} + \left(45 a + 60\right)\cdot 73^{3} + \left(54 a + 49\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 53 + 64\cdot 73 + 4\cdot 73^{2} + 73^{3} + 56\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 35 a + 34 + \left(9 a + 50\right)\cdot 73 + \left(23 a + 37\right)\cdot 73^{2} + \left(27 a + 1\right)\cdot 73^{3} + \left(18 a + 22\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 33 a + 35 + \left(33 a + 22\right)\cdot 73 + \left(59 a + 14\right)\cdot 73^{2} + \left(53 a + 42\right)\cdot 73^{3} + \left(56 a + 24\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 40 a + 61 + \left(39 a + 16\right)\cdot 73 + \left(13 a + 13\right)\cdot 73^{2} + \left(19 a + 71\right)\cdot 73^{3} + \left(16 a + 67\right)\cdot 73^{4} +O(73^{5})\) 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.