Properties

Label 9.190...125.18t272.a.a
Dimension $9$
Group $S_4\wr C_2$
Conductor $1.909\times 10^{13}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $9$
Group: $S_4\wr C_2$
Conductor: \(19087679828125\)\(\medspace = 5^{6} \cdot 1069^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.4.6529513515605.1
Galois orbit size: $1$
Smallest permutation container: 18T272
Parity: even
Determinant: 1.1069.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.4.6529513515605.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} + 11x^{6} - 39x^{5} + 87x^{4} - 178x^{3} + 108x^{2} + 56x + 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 18 a + 25 + \left(24 a + 32\right)\cdot 37 + \left(30 a + 29\right)\cdot 37^{2} + \left(22 a + 26\right)\cdot 37^{3} + \left(3 a + 29\right)\cdot 37^{4} + \left(7 a + 32\right)\cdot 37^{5} + \left(5 a + 22\right)\cdot 37^{6} + \left(9 a + 26\right)\cdot 37^{7} + \left(18 a + 32\right)\cdot 37^{8} + \left(3 a + 31\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 a + 29 + \left(15 a + 35\right)\cdot 37 + \left(14 a + 10\right)\cdot 37^{2} + \left(20 a + 20\right)\cdot 37^{3} + \left(7 a + 6\right)\cdot 37^{4} + \left(7 a + 18\right)\cdot 37^{5} + \left(24 a + 36\right)\cdot 37^{6} + \left(3 a + 17\right)\cdot 37^{7} + \left(29 a + 8\right)\cdot 37^{8} + \left(30 a + 34\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 11\cdot 37 + 11\cdot 37^{2} + 19\cdot 37^{3} + 20\cdot 37^{4} + 36\cdot 37^{6} + 19\cdot 37^{7} + 19\cdot 37^{8} + 5\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 a + 35 + \left(21 a + 3\right)\cdot 37 + \left(22 a + 16\right)\cdot 37^{2} + \left(16 a + 13\right)\cdot 37^{3} + \left(29 a + 16\right)\cdot 37^{4} + \left(29 a + 2\right)\cdot 37^{5} + \left(12 a + 15\right)\cdot 37^{6} + \left(33 a + 8\right)\cdot 37^{7} + \left(7 a + 10\right)\cdot 37^{8} + \left(6 a + 17\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 21 + 30\cdot 37 + 4\cdot 37^{2} + 8\cdot 37^{3} + 12\cdot 37^{4} + 9\cdot 37^{5} + 20\cdot 37^{6} + 23\cdot 37^{7} + 11\cdot 37^{8} + 8\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 12 a + 31 + \left(2 a + 35\right)\cdot 37 + \left(29 a + 8\right)\cdot 37^{2} + \left(30 a + 13\right)\cdot 37^{3} + \left(14 a + 6\right)\cdot 37^{4} + \left(36 a + 22\right)\cdot 37^{5} + \left(4 a + 35\right)\cdot 37^{6} + \left(35 a + 2\right)\cdot 37^{7} + \left(32 a + 10\right)\cdot 37^{8} + \left(23 a + 17\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 25 a + 5 + \left(34 a + 33\right)\cdot 37 + \left(7 a + 11\right)\cdot 37^{2} + \left(6 a + 33\right)\cdot 37^{3} + \left(22 a + 34\right)\cdot 37^{4} + 4\cdot 37^{5} + \left(32 a + 19\right)\cdot 37^{6} + \left(a + 27\right)\cdot 37^{7} + \left(4 a + 32\right)\cdot 37^{8} + \left(13 a + 5\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 19 a + 23 + \left(12 a + 1\right)\cdot 37 + \left(6 a + 17\right)\cdot 37^{2} + \left(14 a + 13\right)\cdot 37^{3} + \left(33 a + 21\right)\cdot 37^{4} + \left(29 a + 20\right)\cdot 37^{5} + \left(31 a + 36\right)\cdot 37^{6} + \left(27 a + 20\right)\cdot 37^{7} + \left(18 a + 22\right)\cdot 37^{8} + \left(33 a + 27\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(3,5,6,7)$
$(1,3)(2,5)(4,6)(7,8)$
$(3,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$9$
$6$$2$$(1,4)(2,8)$$-3$
$9$$2$$(1,4)(2,8)(3,6)(5,7)$$1$
$12$$2$$(3,5)$$-3$
$24$$2$$(1,3)(2,5)(4,6)(7,8)$$3$
$36$$2$$(1,2)(3,5)$$1$
$36$$2$$(1,4)(2,8)(3,5)$$1$
$16$$3$$(3,6,7)$$0$
$64$$3$$(2,4,8)(3,6,7)$$0$
$12$$4$$(1,2,4,8)$$3$
$36$$4$$(1,2,4,8)(3,5,6,7)$$1$
$36$$4$$(1,4)(2,8)(3,5,6,7)$$-1$
$72$$4$$(1,6,4,3)(2,7,8,5)$$-1$
$72$$4$$(1,2,4,8)(3,5)$$-1$
$144$$4$$(1,3,2,5)(4,6)(7,8)$$-1$
$48$$6$$(1,4)(2,8)(3,7,6)$$0$
$96$$6$$(2,8,4)(3,5)$$0$
$192$$6$$(1,5)(2,6,4,7,8,3)$$0$
$144$$8$$(1,5,2,6,4,7,8,3)$$1$
$96$$12$$(1,2,4,8)(3,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.