Normalized defining polynomial
\( x^{14} - 2 x^{13} + 6 x^{11} - 3 x^{10} - 8 x^{9} + 13 x^{8} + 4 x^{7} - 16 x^{6} + 7 x^{5} + 13 x^{4} + \cdots - 1 \)
Invariants
| Degree: | $14$ |
| |
| Signature: | $(2, 6)$ |
| |
| Discriminant: |
\(493856488203125\)
\(\medspace = 5^{7}\cdot 43^{6}\)
|
| |
| Root discriminant: | \(11.21\) |
| |
| Galois root discriminant: | $5^{1/2}43^{1/2}\approx 14.66287829861518$ | ||
| Ramified primes: |
\(5\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}$, $\frac{1}{1235}a^{13}-\frac{84}{1235}a^{12}-\frac{28}{1235}a^{11}-\frac{83}{247}a^{10}+\frac{188}{1235}a^{9}+\frac{137}{1235}a^{8}-\frac{353}{1235}a^{7}+\frac{51}{1235}a^{6}+\frac{1}{1235}a^{5}-\frac{322}{1235}a^{4}-\frac{506}{1235}a^{3}-\frac{3}{247}a^{2}-\frac{9}{1235}a-\frac{491}{1235}$
| 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 $1$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{496}{1235}a^{13}-\frac{909}{1235}a^{12}-\frac{303}{1235}a^{11}+\frac{575}{247}a^{10}-\frac{612}{1235}a^{9}-\frac{4913}{1235}a^{8}+\frac{5222}{1235}a^{7}+\frac{3066}{1235}a^{6}-\frac{8149}{1235}a^{5}+\frac{838}{1235}a^{4}+\frac{7139}{1235}a^{3}-\frac{994}{247}a^{2}-\frac{3229}{1235}a+\frac{2229}{1235}$, $\frac{417}{1235}a^{13}-\frac{448}{1235}a^{12}-\frac{561}{1235}a^{11}+\frac{463}{247}a^{10}+\frac{591}{1235}a^{9}-\frac{3386}{1235}a^{8}+\frac{3469}{1235}a^{7}+\frac{3977}{1235}a^{6}-\frac{4523}{1235}a^{5}+\frac{341}{1235}a^{4}+\frac{6358}{1235}a^{3}-\frac{510}{247}a^{2}-\frac{2518}{1235}a+\frac{1498}{1235}$, $\frac{263}{1235}a^{13}-\frac{109}{1235}a^{12}-\frac{448}{1235}a^{11}+\frac{1017}{1235}a^{10}+\frac{1526}{1235}a^{9}-\frac{1513}{1235}a^{8}+\frac{33}{1235}a^{7}+\frac{4521}{1235}a^{6}-\frac{231}{1235}a^{5}-\frac{2682}{1235}a^{4}+\frac{752}{247}a^{3}+\frac{693}{247}a^{2}-\frac{2367}{1235}a-\frac{188}{247}$, $\frac{131}{1235}a^{13}-\frac{383}{1235}a^{12}+\frac{284}{1235}a^{11}+\frac{469}{1235}a^{10}-\frac{813}{1235}a^{9}-\frac{331}{1235}a^{8}+\frac{1181}{1235}a^{7}-\frac{1223}{1235}a^{6}-\frac{857}{1235}a^{5}+\frac{796}{1235}a^{4}-\frac{265}{247}a^{3}-\frac{393}{247}a^{2}+\frac{56}{1235}a-\frac{119}{247}$, $\frac{47}{247}a^{13}-\frac{474}{1235}a^{12}-\frac{158}{1235}a^{11}+\frac{1769}{1235}a^{10}-\frac{1021}{1235}a^{9}-\frac{2138}{1235}a^{8}+\frac{2754}{1235}a^{7}+\frac{1611}{1235}a^{6}-\frac{4458}{1235}a^{5}+\frac{653}{1235}a^{4}+\frac{2861}{1235}a^{3}-\frac{458}{247}a^{2}-\frac{423}{247}a+\frac{1446}{1235}$, $\frac{54}{65}a^{13}-\frac{18}{13}a^{12}-\frac{6}{13}a^{11}+\frac{314}{65}a^{10}-\frac{79}{65}a^{9}-\frac{83}{13}a^{8}+\frac{542}{65}a^{7}+\frac{62}{13}a^{6}-\frac{674}{65}a^{5}+\frac{35}{13}a^{4}+\frac{587}{65}a^{3}-\frac{71}{13}a^{2}-\frac{226}{65}a+\frac{97}{65}$, $\frac{5}{247}a^{13}-\frac{124}{1235}a^{12}+\frac{782}{1235}a^{11}-\frac{1236}{1235}a^{10}+\frac{254}{1235}a^{9}+\frac{2437}{1235}a^{8}-\frac{921}{1235}a^{7}-\frac{2924}{1235}a^{6}+\frac{5212}{1235}a^{5}+\frac{348}{1235}a^{4}-\frac{3264}{1235}a^{3}+\frac{419}{247}a^{2}+\frac{449}{247}a-\frac{1654}{1235}$
|
| |
| Regulator: | \( 39.9535247267 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 39.9535247267 \cdot 1}{2\cdot\sqrt{493856488203125}}\cr\approx \mathstrut & 0.221240139170 \end{aligned}\]
Galois group
| A solvable group of order 28 |
| The 10 conjugacy class representatives for $D_{14}$ |
| Character table for $D_{14}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 7.1.9938375.1 |
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: | 14.0.4247165798546875.1 |
| 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | R | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(43\)
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *28 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.43.2t1.a.a | $1$ | $ 43 $ | \(\Q(\sqrt{-43}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.215.2t1.a.a | $1$ | $ 5 \cdot 43 $ | \(\Q(\sqrt{-215}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *28 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *28 | 2.215.7t2.a.c | $2$ | $ 5 \cdot 43 $ | 7.1.9938375.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.215.7t2.a.a | $2$ | $ 5 \cdot 43 $ | 7.1.9938375.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.215.14t3.a.b | $2$ | $ 5 \cdot 43 $ | 14.2.493856488203125.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| *28 | 2.215.7t2.a.b | $2$ | $ 5 \cdot 43 $ | 7.1.9938375.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| *28 | 2.215.14t3.a.c | $2$ | $ 5 \cdot 43 $ | 14.2.493856488203125.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| *28 | 2.215.14t3.a.a | $2$ | $ 5 \cdot 43 $ | 14.2.493856488203125.1 | $D_{14}$ (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 *.