Properties

Label 14.19e10_11149e10.42t413.1c1
Dimension 14
Group $S_7$
Conductor $ 19^{10} \cdot 11149^{10}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$14$
Group:$S_7$
Conductor:$181926164762284206907343555768608459562430993045958801= 19^{10} \cdot 11149^{10} $
Artin number field: Splitting field of $f=x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 42T413
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: $x^{2} + 261 x + 5$
Roots: \[ \begin{aligned} r_{ 1 } &= 262977995577 +O\left(263^{ 5 }\right) \\ r_{ 2 } &= 75116772054 a - 225239726357 +O\left(263^{ 5 }\right) \\ r_{ 3 } &= -10241426107 a - 190880459105 +O\left(263^{ 5 }\right) \\ r_{ 4 } &= 2551627499 +O\left(263^{ 5 }\right) \\ r_{ 5 } &= -75116772054 a + 301829928237 +O\left(263^{ 5 }\right) \\ r_{ 6 } &= 10241426107 a - 34436640264 +O\left(263^{ 5 }\right) \\ r_{ 7 } &= -116802725586 +O\left(263^{ 5 }\right) \\ \end{aligned}\]

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.