Normalized defining polynomial
\( x^{12} - x^{11} - 12 x^{10} + 12 x^{9} + 53 x^{8} - 53 x^{7} - 103 x^{6} + 103 x^{5} + 79 x^{4} + \cdots + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[12, 0]$ |
| |
| Discriminant: |
\(1306484927252973\)
\(\medspace = 3^{6}\cdot 13^{11}\)
|
| |
| Root discriminant: | \(18.18\) |
| |
| Galois root discriminant: | $3^{1/2}13^{11/12}\approx 18.18341149267149$ | ||
| Ramified primes: |
\(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{12}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(39=3\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{39}(32,·)$, $\chi_{39}(1,·)$, $\chi_{39}(2,·)$, $\chi_{39}(4,·)$, $\chi_{39}(5,·)$, $\chi_{39}(8,·)$, $\chi_{39}(10,·)$, $\chi_{39}(11,·)$, $\chi_{39}(16,·)$, $\chi_{39}(20,·)$, $\chi_{39}(22,·)$, $\chi_{39}(25,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{6}-6a^{4}+9a^{2}-2$, $a^{3}-3a$, $a^{11}-11a^{9}+a^{8}+44a^{7}-8a^{6}-76a^{5}+20a^{4}+50a^{3}-15a^{2}-6a$, $a^{8}-8a^{6}+a^{5}+20a^{4}-5a^{3}-16a^{2}+5a+1$, $a^{11}-11a^{9}+44a^{7}-a^{6}-77a^{5}+6a^{4}+54a^{3}-8a^{2}-8a$, $a^{10}-11a^{8}+44a^{6}-a^{5}-77a^{4}+6a^{3}+54a^{2}-8a-8$, $a^{4}-4a^{2}+1$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+1$, $a^{9}-a^{8}-9a^{7}+8a^{6}+27a^{5}-20a^{4}-29a^{3}+15a^{2}+6a$, $a^{7}-7a^{5}+14a^{3}-7a-1$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+a^{3}+25a^{2}-3a-1$
|
| |
| Regulator: | \( 2784.79884663 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 2784.79884663 \cdot 1}{2\cdot\sqrt{1306484927252973}}\cr\approx \mathstrut & 0.157787131464 \end{aligned}\]
Galois group
| A cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 3.3.169.1, 4.4.19773.1, \(\Q(\zeta_{13})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | R | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(13\)
| 13.1.12.11a1.1 | $x^{12} + 13$ | $12$ | $1$ | $11$ | $C_{12}$ | $$[\ ]_{12}$$ |