Normalized defining polynomial
\( x^{12} - 2x^{9} + 18x^{8} - 18x^{7} + 14x^{6} - 30x^{5} + 45x^{4} - 52x^{3} + 42x^{2} - 18x + 3 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[0, 6]$ |
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| Discriminant: |
\(11284439629824\)
\(\medspace = 2^{18}\cdot 3^{16}\)
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| Root discriminant: | \(12.24\) |
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| Galois root discriminant: | $2^{3/2}3^{4/3}\approx 12.237893415933033$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $A_4$ |
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| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{2}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{10233}a^{11}+\frac{209}{10233}a^{10}-\frac{662}{10233}a^{9}+\frac{497}{3411}a^{8}+\frac{1549}{3411}a^{7}-\frac{310}{3411}a^{6}+\frac{71}{10233}a^{5}+\frac{4576}{10233}a^{4}-\frac{2062}{10233}a^{3}+\frac{81}{379}a^{2}-\frac{1120}{3411}a+\frac{1273}{3411}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{1818}{379}a^{11}+\frac{6763}{3411}a^{10}+\frac{2071}{3411}a^{9}-\frac{10901}{1137}a^{8}+\frac{93484}{1137}a^{7}-\frac{19729}{379}a^{6}+\frac{48029}{1137}a^{5}-\frac{432514}{3411}a^{4}+\frac{553097}{3411}a^{3}-\frac{203134}{1137}a^{2}+\frac{140956}{1137}a-\frac{35882}{1137}$, $\frac{7795}{10233}a^{11}+\frac{10067}{10233}a^{10}+\frac{8512}{10233}a^{9}-\frac{3055}{3411}a^{8}+\frac{41584}{3411}a^{7}+\frac{8771}{3411}a^{6}+\frac{69083}{10233}a^{5}-\frac{139895}{10233}a^{4}+\frac{127823}{10233}a^{3}-\frac{17491}{1137}a^{2}+\frac{9719}{3411}a+\frac{4984}{3411}$, $\frac{52877}{10233}a^{11}+\frac{12160}{10233}a^{10}-\frac{2029}{10233}a^{9}-\frac{37133}{3411}a^{8}+\frac{307394}{3411}a^{7}-\frac{244196}{3411}a^{6}+\frac{493351}{10233}a^{5}-\frac{1473370}{10233}a^{4}+\frac{1996813}{10233}a^{3}-\frac{243065}{1137}a^{2}+\frac{540754}{3411}a-\frac{154885}{3411}$, $\frac{47557}{10233}a^{11}+\frac{40691}{10233}a^{10}+\frac{28084}{10233}a^{9}-\frac{26278}{3411}a^{8}+\frac{261073}{3411}a^{7}-\frac{58315}{3411}a^{6}+\frac{408977}{10233}a^{5}-\frac{1085666}{10233}a^{4}+\frac{1163456}{10233}a^{3}-\frac{49309}{379}a^{2}+\frac{244517}{3411}a-\frac{32477}{3411}$, $\frac{16381}{10233}a^{11}+\frac{27410}{10233}a^{10}+\frac{27772}{10233}a^{9}-\frac{2974}{3411}a^{8}+\frac{86152}{3411}a^{7}+\frac{52034}{3411}a^{6}+\frac{197738}{10233}a^{5}-\frac{209888}{10233}a^{4}+\frac{184568}{10233}a^{3}-\frac{25447}{1137}a^{2}-\frac{8047}{3411}a+\frac{16351}{3411}$
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| Regulator: | \( 90.7922965712 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 90.7922965712 \cdot 1}{2\cdot\sqrt{11284439629824}}\cr\approx \mathstrut & 0.831492272362 \end{aligned}\]
Galois group
| A solvable group of order 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 4.0.5184.1 x4, 6.2.419904.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 sibling: | 4.0.5184.1 |
| Degree 6 sibling: | 6.2.419904.1 |
| Minimal sibling: | 4.0.5184.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.4.18a4.1 | $x^{12} + 4 x^{10} + 6 x^{9} + 6 x^{8} + 18 x^{7} + 18 x^{6} + 18 x^{5} + 29 x^{4} + 20 x^{3} + 14 x^{2} + 14 x + 7$ | $4$ | $3$ | $18$ | $A_4$ | $$[2, 2]^{3}$$ |
|
\(3\)
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ |
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ | |
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ | |
| 3.1.3.4a2.1 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ |