Properties

Label 14.536...601.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $5.362\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(536\!\cdots\!601\)\(\medspace = 461^{4} \cdot 587^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.270607.1
Galois orbit size: $1$
Smallest permutation container: 21T38
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.270607.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{2} + 58x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 8\cdot 59 + 49\cdot 59^{2} + 43\cdot 59^{3} + 54\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 56 + 58\cdot 59 + 51\cdot 59^{2} + 24\cdot 59^{3} + 21\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 42 a + 14 + \left(29 a + 33\right)\cdot 59 + \left(29 a + 19\right)\cdot 59^{2} + \left(22 a + 6\right)\cdot 59^{3} + \left(14 a + 51\right)\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 54 + 29\cdot 59 + 11\cdot 59^{2} + 28\cdot 59^{3} + 44\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + 56 + \left(29 a + 20\right)\cdot 59 + \left(29 a + 19\right)\cdot 59^{2} + \left(36 a + 58\right)\cdot 59^{3} + \left(44 a + 42\right)\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 11 a + 22 + \left(8 a + 14\right)\cdot 59 + \left(22 a + 35\right)\cdot 59^{2} + \left(34 a + 1\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 48 a + 33 + \left(50 a + 11\right)\cdot 59 + \left(36 a + 49\right)\cdot 59^{2} + \left(24 a + 13\right)\cdot 59^{3} + \left(a + 22\right)\cdot 59^{4} +O(59^{5})\) 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$$(1,2)$$6$
$105$$2$$(1,2)(3,4)(5,6)$$2$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$-1$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$-1$
$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)$$0$

The blue line marks the conjugacy class containing complex conjugation.