Basic invariants
Dimension: | $2$ |
Group: | $D_{7}$ |
Conductor: | \(7203\)\(\medspace = 3 \cdot 7^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.373714754427.1 |
Galois orbit size: | $3$ |
Smallest permutation container: | $D_{7}$ |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $D_7$ |
Projective stem field: | Galois closure of 7.1.373714754427.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} + 7x^{5} - 21x^{4} + 70x^{3} + 273x^{2} + 210x + 900 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 3\cdot 11 + 4\cdot 11^{2} + 7\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\) |
$r_{ 2 }$ | $=$ | \( 3 a + 5 + 8\cdot 11 + \left(9 a + 9\right)\cdot 11^{2} + \left(8 a + 3\right)\cdot 11^{3} + 2\cdot 11^{4} +O(11^{5})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 4 + \left(2 a + 1\right)\cdot 11 + \left(6 a + 1\right)\cdot 11^{2} + \left(2 a + 2\right)\cdot 11^{3} + \left(10 a + 3\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 4 }$ | $=$ | \( 8 a + 8 + \left(a + 9\right)\cdot 11 + \left(8 a + 2\right)\cdot 11^{2} + \left(4 a + 8\right)\cdot 11^{3} + \left(8 a + 9\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 5 }$ | $=$ | \( 3 a + 7 + \left(9 a + 8\right)\cdot 11 + 2 a\cdot 11^{2} + \left(6 a + 8\right)\cdot 11^{3} + \left(2 a + 5\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 6 }$ | $=$ | \( 8 a + 6 + \left(10 a + 6\right)\cdot 11 + \left(a + 1\right)\cdot 11^{2} + \left(2 a + 8\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 7 }$ | $=$ | \( 5 a + 6 + \left(8 a + 5\right)\cdot 11 + \left(4 a + 1\right)\cdot 11^{2} + \left(8 a + 6\right)\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$7$ | $2$ | $(1,5)(2,7)(3,4)$ | $0$ |
$2$ | $7$ | $(1,4,7,6,2,3,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$2$ | $7$ | $(1,7,2,5,4,6,3)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$2$ | $7$ | $(1,6,5,7,3,4,2)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.