Properties

Label 6.23_14369.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 23 \cdot 14369 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$330487= 23 \cdot 14369 $
Artin number field: Splitting field of $f= x^{7} + x^{5} - x^{3} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.23_14369.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 521 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 521 }$: $ x^{2} + 515 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 488 + 5\cdot 521 + 3\cdot 521^{2} + 208\cdot 521^{3} + 195\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 169 a + 518 + \left(14 a + 437\right)\cdot 521 + \left(326 a + 282\right)\cdot 521^{2} + \left(308 a + 342\right)\cdot 521^{3} + \left(289 a + 361\right)\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 155 + 361\cdot 521 + 468\cdot 521^{2} + 423\cdot 521^{3} + 373\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 352 a + 490 + \left(506 a + 354\right)\cdot 521 + \left(194 a + 140\right)\cdot 521^{2} + \left(212 a + 305\right)\cdot 521^{3} + \left(231 a + 227\right)\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 199 + 373\cdot 521 + 481\cdot 521^{2} + 3\cdot 521^{3} + 235\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 327 + 10\cdot 521 + 31\cdot 521^{2} + 419\cdot 521^{3} + 143\cdot 521^{4} +O\left(521^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 428 + 18\cdot 521 + 155\cdot 521^{2} + 381\cdot 521^{3} + 25\cdot 521^{4} +O\left(521^{ 5 }\right)$

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$$1$$()$$6$
$21$$2$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-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$$(1,2,3,4,5)(6,7)$$-1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$
The blue line marks the conjugacy class containing complex conjugation.