Properties

Label 15.801...151.42t412.a.a
Dimension $15$
Group $S_7$
Conductor $8.011\times 10^{27}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $15$
Group: $S_7$
Conductor: \(801\!\cdots\!151\)\(\medspace = 11^{5} \cdot 89^{5} \cdot 389^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.380831.1
Galois orbit size: $1$
Smallest permutation container: 42T412
Parity: odd
Determinant: 1.380831.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.380831.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 a + 16 + \left(10 a + 30\right)\cdot 43 + \left(38 a + 23\right)\cdot 43^{2} + 16\cdot 43^{3} + \left(3 a + 24\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 32 a + 16 + \left(7 a + 36\right)\cdot 43 + \left(24 a + 36\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + \left(22 a + 17\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 5 + \left(35 a + 12\right)\cdot 43 + \left(18 a + 10\right)\cdot 43^{2} + \left(9 a + 26\right)\cdot 43^{3} + \left(20 a + 6\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 8\cdot 43 + 14\cdot 43^{2} + 10\cdot 43^{3} +O(43^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 6 a + \left(4 a + 34\right)\cdot 43 + \left(25 a + 28\right)\cdot 43^{2} + \left(29 a + 37\right)\cdot 43^{3} + \left(23 a + 29\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 37 a + 6 + \left(38 a + 32\right)\cdot 43 + \left(17 a + 6\right)\cdot 43^{2} + \left(13 a + 42\right)\cdot 43^{3} + \left(19 a + 23\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 21 a + 38 + \left(32 a + 18\right)\cdot 43 + \left(4 a + 8\right)\cdot 43^{2} + \left(42 a + 22\right)\cdot 43^{3} + \left(39 a + 26\right)\cdot 43^{4} +O(43^{5})\) 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$$1$$()$$15$
$21$$2$$(1,2)$$5$
$105$$2$$(1,2)(3,4)(5,6)$$-3$
$105$$2$$(1,2)(3,4)$$-1$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$1$
$630$$4$$(1,2,3,4)(5,6)$$-1$
$504$$5$$(1,2,3,4,5)$$0$
$210$$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$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$1$

The blue line marks the conjugacy class containing complex conjugation.