Basic invariants
Dimension: | $2$ |
Group: | $D_{7}$ |
Conductor: | \(215\)\(\medspace = 5 \cdot 43 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.9938375.1 |
Galois orbit size: | $3$ |
Smallest permutation container: | $D_{7}$ |
Parity: | odd |
Determinant: | 1.215.2t1.a.a |
Projective image: | $D_7$ |
Projective stem field: | Galois closure of 7.1.9938375.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} + 4x^{4} - 6x^{3} + x^{2} + 4x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a + \left(7 a + 6\right)\cdot 13 + \left(10 a + 2\right)\cdot 13^{2} + \left(12 a + 8\right)\cdot 13^{3} + \left(7 a + 5\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$ |
$=$ |
\( 12 a + 1 + \left(5 a + 12\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + 10\cdot 13^{3} + 5 a\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 9 a + 5 + \left(12 a + 11\right)\cdot 13 + \left(6 a + 2\right)\cdot 13^{2} + 5 a\cdot 13^{3} + \left(3 a + 10\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 + 11\cdot 13 + 10\cdot 13^{2} + 3\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a + 1 + 2\cdot 13 + \left(6 a + 10\right)\cdot 13^{2} + \left(7 a + 11\right)\cdot 13^{3} + \left(9 a + 7\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 8 a + 11 + \left(9 a + 3\right)\cdot 13 + \left(11 a + 2\right)\cdot 13^{2} + \left(6 a + 11\right)\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 5 a + 6 + \left(3 a + 5\right)\cdot 13 + \left(a + 4\right)\cdot 13^{2} + \left(6 a + 6\right)\cdot 13^{3} + \left(12 a + 9\right)\cdot 13^{4} +O(13^{5})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$7$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
$2$ | $7$ | $(1,6,5,4,3,7,2)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$2$ | $7$ | $(1,5,3,2,6,4,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$2$ | $7$ | $(1,4,2,5,7,6,3)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.