# Properties

 Label 2.151.7t2.a Dimension $2$ Group $D_{7}$ Conductor $151$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{7}$ Conductor: $$151$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 7.1.3442951.1 Galois orbit size: $3$ Smallest permutation container: $D_{7}$ Parity: odd Projective image: $D_7$ Projective field: 7.1.3442951.1

## Galois action

### Roots of defining polynomial

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

### Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

 Cycle notation $(1,4)(2,5)(3,6)$ $(1,6)(3,5)(4,7)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $c3$ $1$ $1$ $()$ $2$ $2$ $2$ $7$ $2$ $(1,4)(2,5)(3,6)$ $0$ $0$ $0$ $2$ $7$ $(1,7,4,6,5,2,3)$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $2$ $7$ $(1,4,5,3,7,6,2)$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $2$ $7$ $(1,6,3,4,2,7,5)$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$
The blue line marks the conjugacy class containing complex conjugation.