Properties

Label 5.31709.6t16.a
Dimension $5$
Group $S_6$
Conductor $31709$
Indicator $1$

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:\(31709\)\(\medspace = 37 \cdot 857 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.31709.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: even
Projective image: $S_6$
Projective field: Galois closure of 6.2.31709.1

Galois action

Roots of defining polynomial

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:
$r_{ 1 }$ $=$ \( 39 a + 28 + \left(49 a + 25\right)\cdot 73 + \left(2 a + 71\right)\cdot 73^{2} + \left(48 a + 20\right)\cdot 73^{3} + \left(55 a + 50\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 40 a + 28 + \left(67 a + 43\right)\cdot 73 + 37\cdot 73^{2} + \left(38 a + 30\right)\cdot 73^{3} + \left(49 a + 25\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 34 a + 72 + \left(23 a + 61\right)\cdot 73 + \left(70 a + 29\right)\cdot 73^{2} + \left(24 a + 16\right)\cdot 73^{3} + \left(17 a + 23\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 71 + 6\cdot 73 + 42\cdot 73^{2} + 36\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 33 a + 2 + \left(5 a + 60\right)\cdot 73 + \left(72 a + 45\right)\cdot 73^{2} + \left(34 a + 70\right)\cdot 73^{3} + \left(23 a + 62\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 19 + 21\cdot 73 + 65\cdot 73^{2} + 43\cdot 73^{3} + 22\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-1$
$15$ $2$ $(1,2)$ $3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $-1$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.