Normalized defining polynomial
\( x^{14} - 7 x^{13} + 22 x^{12} - 41 x^{11} + 61 x^{10} - 96 x^{9} + 156 x^{8} - 213 x^{7} + 230 x^{6} + \cdots - 9 \)
Invariants
| Degree: | $14$ |
| |
| Signature: | $[2, 6]$ |
| |
| Discriminant: |
\(4466153794705739173\)
\(\medspace = 19^{6}\cdot 37^{7}\)
|
| |
| Root discriminant: | \(21.49\) |
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| Galois root discriminant: | $19^{1/2}37^{1/2}\approx 26.514147167125703$ | ||
| Ramified primes: |
\(19\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{15}a^{11}+\frac{2}{15}a^{10}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{4}{15}a^{3}+\frac{7}{15}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{255}a^{12}-\frac{2}{85}a^{11}-\frac{41}{255}a^{10}+\frac{1}{51}a^{9}+\frac{36}{85}a^{8}-\frac{6}{85}a^{7}+\frac{2}{5}a^{6}-\frac{7}{17}a^{5}-\frac{79}{255}a^{4}+\frac{33}{85}a^{3}-\frac{124}{255}a^{2}+\frac{58}{255}a-\frac{9}{85}$, $\frac{1}{765}a^{13}+\frac{1}{765}a^{12}-\frac{1}{51}a^{11}+\frac{109}{765}a^{10}-\frac{3}{85}a^{9}-\frac{3}{85}a^{8}-\frac{22}{51}a^{7}+\frac{28}{85}a^{6}-\frac{49}{765}a^{5}+\frac{311}{765}a^{4}+\frac{14}{255}a^{3}-\frac{334}{765}a^{2}+\frac{23}{51}a-\frac{38}{85}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{179}{765}a^{13}-\frac{877}{765}a^{12}+\frac{503}{255}a^{11}-\frac{637}{765}a^{10}+\frac{199}{255}a^{9}-\frac{22}{5}a^{8}+\frac{1334}{255}a^{7}+\frac{116}{85}a^{6}-\frac{5756}{765}a^{5}+\frac{7513}{765}a^{4}-\frac{3221}{255}a^{3}+\frac{4399}{765}a^{2}-\frac{239}{255}a-\frac{251}{85}$, $\frac{47}{153}a^{13}-\frac{1472}{765}a^{12}+\frac{446}{85}a^{11}-\frac{6194}{765}a^{10}+\frac{185}{17}a^{9}-\frac{1554}{85}a^{8}+\frac{7699}{255}a^{7}-\frac{3093}{85}a^{6}+\frac{4933}{153}a^{5}-\frac{16972}{765}a^{4}+\frac{521}{85}a^{3}+\frac{106}{45}a^{2}-\frac{1319}{255}a+\frac{118}{85}$, $\frac{47}{153}a^{13}-\frac{1583}{765}a^{12}+\frac{104}{17}a^{11}-\frac{8018}{765}a^{10}+\frac{252}{17}a^{9}-\frac{2036}{85}a^{8}+\frac{1979}{51}a^{7}-\frac{4266}{85}a^{6}+\frac{7723}{153}a^{5}-\frac{30388}{765}a^{4}+\frac{302}{17}a^{3}+\frac{521}{765}a^{2}-\frac{370}{51}a+\frac{281}{85}$, $\frac{218}{765}a^{13}-\frac{1522}{765}a^{12}+\frac{308}{51}a^{11}-\frac{7918}{765}a^{10}+\frac{209}{15}a^{9}-\frac{1899}{85}a^{8}+\frac{1933}{51}a^{7}-\frac{4181}{85}a^{6}+\frac{35848}{765}a^{5}-\frac{1561}{45}a^{4}+\frac{4112}{255}a^{3}-\frac{362}{765}a^{2}-\frac{286}{51}a+\frac{251}{85}$, $\frac{52}{45}a^{13}-\frac{5206}{765}a^{12}+\frac{285}{17}a^{11}-\frac{16819}{765}a^{10}+\frac{2408}{85}a^{9}-\frac{4587}{85}a^{8}+\frac{4435}{51}a^{7}-\frac{449}{5}a^{6}+\frac{52219}{765}a^{5}-\frac{31916}{765}a^{4}-\frac{78}{85}a^{3}+\frac{11104}{765}a^{2}-\frac{581}{51}a-\frac{362}{85}$, $\frac{48}{85}a^{13}-\frac{203}{51}a^{12}+\frac{61}{5}a^{11}-\frac{5431}{255}a^{10}+\frac{7487}{255}a^{9}-\frac{801}{17}a^{8}+\frac{6686}{85}a^{7}-\frac{8746}{85}a^{6}+\frac{8633}{85}a^{5}-\frac{4045}{51}a^{4}+\frac{3328}{85}a^{3}-\frac{566}{255}a^{2}-\frac{2888}{255}a+\frac{696}{85}$, $\frac{48}{85}a^{13}-\frac{857}{255}a^{12}+\frac{721}{85}a^{11}-\frac{2984}{255}a^{10}+\frac{1314}{85}a^{9}-\frac{141}{5}a^{8}+\frac{3868}{85}a^{7}-\frac{4224}{85}a^{6}+\frac{3443}{85}a^{5}-\frac{6697}{255}a^{4}+\frac{297}{85}a^{3}+\frac{1262}{255}a^{2}-\frac{543}{85}a-\frac{131}{85}$
|
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| Regulator: | \( 13575.407460757579 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 13575.407460757579 \cdot 1}{2\cdot\sqrt{4466153794705739173}}\cr\approx \mathstrut & 0.790487928739484 \end{aligned}\]
Galois group
| A solvable group of order 28 |
| The 10 conjugacy class representatives for $D_{14}$ |
| Character table for $D_{14}$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), 7.1.347428927.1 |
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: | 14.0.2293430327011055251.1 |
| Minimal sibling: | 14.0.2293430327011055251.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.2.0.1}{2} }^{6}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{7}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.14.0.1}{14} }$ | R | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(37\)
| 37.1.2.1a1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 37.2.2.2a1.2 | $x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 37.2.2.2a1.2 | $x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 37.2.2.2a1.2 | $x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
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 | 1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.703.2t1.a.a | $1$ | $ 19 \cdot 37 $ | \(\Q(\sqrt{-703}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *28 | 2.703.14t3.a.b | $2$ | $ 19 \cdot 37 $ | 14.2.4466153794705739173.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| *28 | 2.703.7t2.a.c | $2$ | $ 19 \cdot 37 $ | 7.1.347428927.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.703.7t2.a.b | $2$ | $ 19 \cdot 37 $ | 7.1.347428927.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.703.7t2.a.a | $2$ | $ 19 \cdot 37 $ | 7.1.347428927.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.703.14t3.a.a | $2$ | $ 19 \cdot 37 $ | 14.2.4466153794705739173.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| *28 | 2.703.14t3.a.c | $2$ | $ 19 \cdot 37 $ | 14.2.4466153794705739173.1 | $D_{14}$ (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 *.