Properties

Label 4.5e3_61e4.6t10.2c1
Dimension 4
Group $C_3^2:C_4$
Conductor $ 5^{3} \cdot 61^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:C_4$
Conductor:$1730730125= 5^{3} \cdot 61^{4} $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 6 x^{4} - 2 x^{3} + 4 x^{2} + 9 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:C_4$
Parity: Even
Determinant: 1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 4 + 13\cdot 19 + 15\cdot 19^{2} + 12\cdot 19^{3} + 8\cdot 19^{4} + 12\cdot 19^{5} + 9\cdot 19^{6} + 4\cdot 19^{7} + 10\cdot 19^{8} + 12\cdot 19^{9} + 19^{10} + 2\cdot 19^{11} + 17\cdot 19^{12} + 16\cdot 19^{13} + 19^{14} +O\left(19^{ 15 }\right)$
$r_{ 2 }$ $=$ $ 16 + 17\cdot 19 + 5\cdot 19^{2} + 18\cdot 19^{3} + 6\cdot 19^{4} + 17\cdot 19^{5} + 9\cdot 19^{6} + 18\cdot 19^{7} + 18\cdot 19^{8} + 17\cdot 19^{9} + 5\cdot 19^{13} + 11\cdot 19^{14} +O\left(19^{ 15 }\right)$
$r_{ 3 }$ $=$ $ 17 a + 5 + \left(7 a + 6\right)\cdot 19 + \left(9 a + 1\right)\cdot 19^{2} + \left(7 a + 18\right)\cdot 19^{3} + \left(2 a + 2\right)\cdot 19^{4} + \left(7 a + 5\right)\cdot 19^{5} + \left(3 a + 9\right)\cdot 19^{6} + \left(6 a + 14\right)\cdot 19^{7} + \left(a + 6\right)\cdot 19^{8} + \left(8 a + 18\right)\cdot 19^{9} + \left(10 a + 14\right)\cdot 19^{10} + \left(14 a + 10\right)\cdot 19^{11} + \left(17 a + 10\right)\cdot 19^{12} + \left(13 a + 15\right)\cdot 19^{13} + \left(9 a + 4\right)\cdot 19^{14} +O\left(19^{ 15 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 2 + \left(3 a + 13\right)\cdot 19 + \left(3 a + 15\right)\cdot 19^{2} + \left(10 a + 1\right)\cdot 19^{3} + \left(11 a + 10\right)\cdot 19^{4} + \left(17 a + 12\right)\cdot 19^{5} + \left(10 a + 14\right)\cdot 19^{6} + \left(10 a + 10\right)\cdot 19^{7} + \left(2 a + 13\right)\cdot 19^{8} + \left(18 a + 2\right)\cdot 19^{9} + \left(15 a + 12\right)\cdot 19^{10} + \left(3 a + 1\right)\cdot 19^{11} + \left(14 a + 16\right)\cdot 19^{13} + \left(a + 15\right)\cdot 19^{14} +O\left(19^{ 15 }\right)$
$r_{ 5 }$ $=$ $ 12 a + 9 + \left(15 a + 9\right)\cdot 19 + \left(15 a + 15\right)\cdot 19^{2} + \left(8 a + 8\right)\cdot 19^{3} + \left(7 a + 11\right)\cdot 19^{4} + \left(a + 18\right)\cdot 19^{5} + \left(8 a + 7\right)\cdot 19^{6} + \left(8 a + 10\right)\cdot 19^{7} + \left(16 a + 5\right)\cdot 19^{8} + 18\cdot 19^{9} + \left(3 a + 9\right)\cdot 19^{10} + \left(15 a + 8\right)\cdot 19^{11} + \left(18 a + 15\right)\cdot 19^{12} + \left(4 a + 10\right)\cdot 19^{13} + \left(17 a + 3\right)\cdot 19^{14} +O\left(19^{ 15 }\right)$
$r_{ 6 }$ $=$ $ 2 a + 3 + \left(11 a + 16\right)\cdot 19 + \left(9 a + 2\right)\cdot 19^{2} + \left(11 a + 16\right)\cdot 19^{3} + \left(16 a + 16\right)\cdot 19^{4} + \left(11 a + 9\right)\cdot 19^{5} + \left(15 a + 5\right)\cdot 19^{6} + \left(12 a + 17\right)\cdot 19^{7} + \left(17 a + 1\right)\cdot 19^{8} + \left(10 a + 6\right)\cdot 19^{9} + \left(8 a + 17\right)\cdot 19^{10} + \left(4 a + 14\right)\cdot 19^{11} + \left(a + 13\right)\cdot 19^{12} + \left(5 a + 11\right)\cdot 19^{13} + 9 a\cdot 19^{14} +O\left(19^{ 15 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$9$$2$$(1,4)(2,3)$$0$
$4$$3$$(1,4,5)$$-2$
$4$$3$$(1,4,5)(2,3,6)$$1$
$9$$4$$(1,2,4,3)(5,6)$$0$
$9$$4$$(1,3,4,2)(5,6)$$0$
The blue line marks the conjugacy class containing complex conjugation.