Properties

Label 1.3_5_7.2t1.1c1
Dimension 1
Group $C_2$
Conductor $ 3 \cdot 5 \cdot 7 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$1$
Group:$C_2$
Conductor:$105= 3 \cdot 5 \cdot 7 $
Artin number field: Splitting field of $f= x^{2} - x - 26 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2$
Parity: Even
Corresponding Dirichlet character: \(\displaystyle\left(\frac{105}{\bullet}\right)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11\cdot 13 + 3\cdot 13^{2} + 10\cdot 13^{3} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 1 + 2\cdot 13 + 9\cdot 13^{2} + 2\cdot 13^{3} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$

Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $

Cycle notation
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 } $ Character value
$1$$1$$()$$1$
$1$$2$$(1,2)$$-1$
The blue line marks the conjugacy class containing complex conjugation.