Properties

Label 6.334...151.14t46.a.a
Dimension $6$
Group $S_7$
Conductor $3.347\times 10^{27}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_7$
Conductor: \(334\!\cdots\!151\)\(\medspace = 319831^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.319831.1
Galois orbit size: $1$
Smallest permutation container: 14T46
Parity: odd
Determinant: 1.319831.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.319831.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} + 2x^{4} - 3x^{3} + 4x^{2} - 3x + 1 \) 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:
$r_{ 1 }$ $=$ \( 33 a + 28 + \left(10 a + 67\right)\cdot 73 + \left(47 a + 5\right)\cdot 73^{2} + \left(29 a + 43\right)\cdot 73^{3} + \left(28 a + 23\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 35 + 12\cdot 73 + 24\cdot 73^{2} + 56\cdot 73^{3} + 64\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 64 a + \left(69 a + 38\right)\cdot 73 + \left(18 a + 38\right)\cdot 73^{2} + \left(24 a + 42\right)\cdot 73^{3} + \left(51 a + 63\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 a + 46 + \left(3 a + 37\right)\cdot 73 + \left(54 a + 25\right)\cdot 73^{2} + \left(48 a + 23\right)\cdot 73^{3} + \left(21 a + 47\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 + 8\cdot 73 + 41\cdot 73^{2} + 8\cdot 73^{3} + 69\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 40 a + 54 + \left(62 a + 65\right)\cdot 73 + \left(25 a + 63\right)\cdot 73^{2} + \left(43 a + 11\right)\cdot 73^{3} + \left(44 a + 6\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 44 + 62\cdot 73 + 19\cdot 73^{2} + 33\cdot 73^{3} + 17\cdot 73^{4} +O(73^{5})\) 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$$1$$()$$6$
$21$$2$$(1,2)$$-4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$-2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$-1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$1$
$420$$12$$(1,2,3,4)(5,6,7)$$1$

The blue line marks the conjugacy class containing complex conjugation.