Properties

Label 14.150...351.30t565.a.a
Dimension $14$
Group $S_7$
Conductor $1.507\times 10^{27}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(150\!\cdots\!351\)\(\medspace = 7^{5} \cdot 38953^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.272671.1
Galois orbit size: $1$
Smallest permutation container: 30T565
Parity: odd
Determinant: 1.272671.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.272671.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{2} + 69x + 7 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 23 a + 54 + \left(64 a + 36\right)\cdot 71 + \left(6 a + 68\right)\cdot 71^{2} + \left(5 a + 16\right)\cdot 71^{3} + \left(41 a + 27\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 27 a + 69 + \left(55 a + 31\right)\cdot 71 + \left(19 a + 34\right)\cdot 71^{2} + \left(15 a + 10\right)\cdot 71^{3} + \left(52 a + 26\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 58 + 9\cdot 71 + 60\cdot 71^{2} + 7\cdot 71^{3} + 39\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 56 + 25\cdot 71 + 32\cdot 71^{2} + 12\cdot 71^{3} + 58\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 48 a + 29 + 6 a\cdot 71 + \left(64 a + 18\right)\cdot 71^{2} + \left(65 a + 20\right)\cdot 71^{3} + \left(29 a + 33\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 44 a + 52 + \left(15 a + 44\right)\cdot 71 + \left(51 a + 18\right)\cdot 71^{2} + \left(55 a + 21\right)\cdot 71^{3} + \left(18 a + 44\right)\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 39 + 63\cdot 71 + 51\cdot 71^{2} + 52\cdot 71^{3} + 55\cdot 71^{4} +O(71^{5})\) 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$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$-1$
$280$$3$$(1,2,3)(4,5,6)$$2$
$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)$$0$
$504$$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.