Normalized defining polynomial
\( x^{12} + 4x^{10} + 9x^{8} + 16x^{6} + 22x^{4} + 20x^{2} + 9 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $(0, 6)$ |
| |
| Discriminant: |
\(629407744000000\)
\(\medspace = 2^{24}\cdot 5^{6}\cdot 7^{4}\)
|
| |
| Root discriminant: | \(17.11\) |
| |
| Galois root discriminant: | $2^{2}5^{1/2}7^{2/3}\approx 32.72982527225799$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_6$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}, \sqrt{-5})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{3}a^{10}+\frac{4}{3}a^{8}+\frac{7}{3}a^{6}+4a^{4}+\frac{11}{3}a^{2}+1$, $a^{2}+1$, $\frac{2}{3}a^{11}-\frac{1}{3}a^{10}+\frac{7}{3}a^{9}-\frac{2}{3}a^{8}+\frac{13}{3}a^{7}-\frac{2}{3}a^{6}+7a^{5}-2a^{4}+\frac{26}{3}a^{3}-\frac{4}{3}a^{2}+4a+1$, $\frac{1}{3}a^{11}+\frac{2}{3}a^{10}+\frac{4}{3}a^{9}+\frac{4}{3}a^{8}+3a^{7}+\frac{7}{3}a^{6}+\frac{16}{3}a^{5}+4a^{4}+\frac{19}{3}a^{3}+\frac{14}{3}a^{2}+\frac{14}{3}a+3$, $\frac{5}{3}a^{11}+\frac{1}{3}a^{10}+\frac{13}{3}a^{9}+\frac{2}{3}a^{8}+\frac{25}{3}a^{7}+\frac{5}{3}a^{6}+14a^{5}+2a^{4}+\frac{47}{3}a^{3}+\frac{10}{3}a^{2}+9a+1$
|
| |
| Regulator: | \( 783.4634695325174 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 783.4634695325174 \cdot 2}{2\cdot\sqrt{629407744000000}}\cr\approx \mathstrut & 1.92146253359586 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-2}, \sqrt{-5})\), 6.0.392000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 36 |
| Degree 18 siblings: | 18.6.232218265089212416000000000.1, 18.0.1857746120713699328000000.1 |
| 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 | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{6}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.4.24b1.9 | $x^{12} + 4 x^{10} + 8 x^{9} + 6 x^{8} + 24 x^{7} + 24 x^{6} + 24 x^{5} + 41 x^{4} + 32 x^{3} + 20 x^{2} + 24 x + 17$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $$[2, 3]^{3}$$ |
|
\(5\)
| 5.6.2.6a1.2 | $x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $$[\ ]_{2}^{6}$$ |
|
\(7\)
| 7.6.1.0a1.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |
| 7.2.3.4a1.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 169 x + 55$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *36 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *36 | 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *36 | 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.280.6t1.a.a | $1$ | $ 2^{3} \cdot 5 \cdot 7 $ | 6.6.153664000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.56.6t1.c.a | $1$ | $ 2^{3} \cdot 7 $ | 6.0.1229312.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.140.6t1.b.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.19208000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.140.6t1.b.b | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.19208000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.280.6t1.a.b | $1$ | $ 2^{3} \cdot 5 \cdot 7 $ | 6.6.153664000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.56.6t1.c.b | $1$ | $ 2^{3} \cdot 7 $ | 6.0.1229312.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.980.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 3.1.980.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.15680.6t3.e.a | $2$ | $ 2^{6} \cdot 5 \cdot 7^{2}$ | 6.0.122931200.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *36 | 2.140.6t5.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.392000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.2240.12t18.a.a | $2$ | $ 2^{6} \cdot 5 \cdot 7 $ | 12.0.629407744000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.2240.12t18.a.b | $2$ | $ 2^{6} \cdot 5 \cdot 7 $ | 12.0.629407744000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.140.6t5.a.b | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.0.392000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.