Normalized defining polynomial
\( x^{14} - x^{13} + 2 x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 3 x^{8} - 4 x^{7} + 2 x^{6} - 4 x^{5} + \cdots + 1 \)
Invariants
| Degree: | $14$ |
| |
| Signature: | $(0, 7)$ |
| |
| Discriminant: |
\(-280155320935227\)
\(\medspace = -\,3^{7}\cdot 71^{6}\)
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| |
| Root discriminant: | \(10.76\) |
| |
| Galois root discriminant: | $3^{1/2}71^{1/2}\approx 14.594519519326424$ | ||
| Ramified primes: |
\(3\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{3}{7}a^{11}+\frac{2}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}$, $\frac{1}{119}a^{13}+\frac{2}{119}a^{12}-\frac{43}{119}a^{11}-\frac{45}{119}a^{10}+\frac{4}{119}a^{9}+\frac{59}{119}a^{8}+\frac{27}{119}a^{7}+\frac{9}{119}a^{6}+\frac{12}{119}a^{5}+\frac{32}{119}a^{4}-\frac{38}{119}a^{3}+\frac{22}{119}a^{2}+\frac{3}{119}a+\frac{11}{119}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $6$ |
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| Torsion generator: |
\( -\frac{47}{119} a^{13} + \frac{6}{17} a^{12} - \frac{10}{17} a^{11} + \frac{1}{17} a^{10} - \frac{86}{119} a^{9} + \frac{134}{119} a^{8} - \frac{96}{119} a^{7} + \frac{138}{119} a^{6} - \frac{88}{119} a^{5} + \frac{128}{119} a^{4} - \frac{12}{17} a^{3} + \frac{20}{119} a^{2} - \frac{260}{119} a + \frac{10}{119} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{73}{119}a^{13}-\frac{18}{17}a^{12}+\frac{30}{17}a^{11}-\frac{20}{17}a^{10}+\frac{207}{119}a^{9}-\frac{317}{119}a^{8}+\frac{339}{119}a^{7}-\frac{346}{119}a^{6}+\frac{162}{119}a^{5}-\frac{214}{119}a^{4}+\frac{19}{17}a^{3}-\frac{26}{119}a^{2}+\frac{219}{119}a-\frac{13}{119}$, $a$, $\frac{43}{119}a^{13}-\frac{12}{17}a^{12}+\frac{20}{17}a^{11}-\frac{19}{17}a^{10}+\frac{223}{119}a^{9}-\frac{353}{119}a^{8}+\frac{379}{119}a^{7}-\frac{344}{119}a^{6}+\frac{278}{119}a^{5}-\frac{307}{119}a^{4}+\frac{41}{17}a^{3}-\frac{74}{119}a^{2}+\frac{248}{119}a-\frac{37}{119}$, $\frac{73}{119}a^{13}-\frac{18}{17}a^{12}+\frac{30}{17}a^{11}-\frac{20}{17}a^{10}+\frac{207}{119}a^{9}-\frac{317}{119}a^{8}+\frac{339}{119}a^{7}-\frac{346}{119}a^{6}+\frac{162}{119}a^{5}-\frac{214}{119}a^{4}+\frac{19}{17}a^{3}-\frac{26}{119}a^{2}+\frac{219}{119}a+\frac{106}{119}$, $\frac{10}{17}a^{13}-\frac{14}{17}a^{12}+\frac{29}{17}a^{11}-\frac{25}{17}a^{10}+\frac{40}{17}a^{9}-\frac{56}{17}a^{8}+\frac{49}{17}a^{7}-\frac{63}{17}a^{6}+\frac{35}{17}a^{5}-\frac{37}{17}a^{4}+\frac{28}{17}a^{3}-\frac{1}{17}a^{2}+\frac{47}{17}a+\frac{8}{17}$, $\frac{57}{119}a^{13}-\frac{39}{119}a^{12}+\frac{65}{119}a^{11}-\frac{15}{119}a^{10}+\frac{143}{119}a^{9}-\frac{190}{119}a^{8}+\frac{26}{119}a^{7}-\frac{19}{17}a^{6}+\frac{89}{119}a^{5}-\frac{250}{119}a^{4}+\frac{27}{119}a^{3}+\frac{14}{17}a^{2}+\frac{290}{119}a+\frac{7}{17}$
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| Regulator: | \( 64.9438034848 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 64.9438034848 \cdot 1}{6\cdot\sqrt{280155320935227}}\cr\approx \mathstrut & 0.250003512392 \end{aligned}\]
Galois group
| A solvable group of order 28 |
| The 10 conjugacy class representatives for $D_{14}$ |
| Character table for $D_{14}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 28 |
| Degree 14 sibling: | 14.2.19891027786401117.1 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.14.0.1}{14} }$ | R | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.7.2.7a1.2 | $x^{14} + 4 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{2} + 4$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ |
|
\(71\)
| 71.2.1.0a1.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 71.2.2.2a1.2 | $x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 71.2.2.2a1.2 | $x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 71.2.2.2a1.2 | $x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *28 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *28 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.213.2t1.a.a | $1$ | $ 3 \cdot 71 $ | \(\Q(\sqrt{213}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *28 | 2.71.7t2.a.c | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.71.7t2.a.a | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.639.14t3.a.a | $2$ | $ 3^{2} \cdot 71 $ | 14.0.280155320935227.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| *28 | 2.71.7t2.a.b | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.639.14t3.a.b | $2$ | $ 3^{2} \cdot 71 $ | 14.0.280155320935227.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| *28 | 2.639.14t3.a.c | $2$ | $ 3^{2} \cdot 71 $ | 14.0.280155320935227.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.