Normalized defining polynomial
\( x^{12} - 10x^{10} + 17x^{8} + 42x^{6} + 46x^{4} - 8x^{2} + 1 \)
Invariants
| Degree: | $12$ |
| |
| Signature: | $[0, 6]$ |
| |
| Discriminant: |
\(34162868224000000\)
\(\medspace = 2^{24}\cdot 5^{6}\cdot 19^{4}\)
|
| |
| Root discriminant: | \(23.87\) |
| |
| Galois root discriminant: | $2^{2}5^{1/2}19^{2/3}\approx 63.686501757102$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
|
| |
| 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: | \(\Q(i, \sqrt{10})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{524}a^{10}+\frac{27}{131}a^{8}-\frac{1}{4}a^{7}+\frac{27}{262}a^{6}+\frac{1}{4}a^{5}+\frac{63}{262}a^{4}-\frac{151}{524}a^{2}-\frac{1}{4}a-\frac{141}{524}$, $\frac{1}{524}a^{11}-\frac{23}{524}a^{9}-\frac{77}{524}a^{7}-\frac{1}{4}a^{6}-\frac{34}{131}a^{5}+\frac{1}{4}a^{4}+\frac{121}{262}a^{3}+\frac{121}{524}a-\frac{1}{4}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
\( \frac{50}{131} a^{11} - \frac{495}{131} a^{9} + \frac{1601}{262} a^{7} + \frac{4347}{262} a^{5} + \frac{2537}{131} a^{3} - \frac{83}{262} a \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{5}{131}a^{10}-\frac{99}{262}a^{8}+\frac{147}{262}a^{6}+\frac{237}{131}a^{4}+\frac{717}{262}a^{2}-\frac{50}{131}$, $\frac{21}{262}a^{10}-\frac{221}{262}a^{8}+\frac{479}{262}a^{6}+\frac{275}{131}a^{4}+\frac{445}{131}a^{2}-\frac{79}{262}$, $\frac{831}{524}a^{11}-\frac{68}{131}a^{10}-\frac{2060}{131}a^{9}+\frac{2719}{524}a^{8}+\frac{6717}{262}a^{7}-\frac{4601}{524}a^{6}+\frac{18031}{262}a^{5}-\frac{5739}{262}a^{4}+\frac{41151}{524}a^{3}-\frac{12769}{524}a^{2}-\frac{3463}{524}a+\frac{967}{262}$, $\frac{32}{131}a^{11}-\frac{45}{262}a^{10}-\frac{1241}{524}a^{9}+\frac{891}{524}a^{8}+\frac{1803}{524}a^{7}-\frac{727}{262}a^{6}+\frac{2955}{262}a^{5}-\frac{3873}{524}a^{4}+\frac{7527}{524}a^{3}-\frac{4095}{524}a^{2}+\frac{539}{262}a+\frac{507}{524}$, $\frac{20}{131}a^{11}-\frac{73}{524}a^{10}-\frac{198}{131}a^{9}+\frac{381}{262}a^{8}+\frac{1307}{524}a^{7}-\frac{396}{131}a^{6}+\frac{3137}{524}a^{5}-\frac{1193}{262}a^{4}+\frac{1303}{131}a^{3}-\frac{1815}{524}a^{2}-\frac{145}{524}a+\frac{337}{524}$
|
| |
| Regulator: | \( 1054.7017720420233 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1054.7017720420233 \cdot 2}{4\cdot\sqrt{34162868224000000}}\cr\approx \mathstrut & 0.175550361164461 \end{aligned}\]
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
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{10}) \), \(\Q(i, \sqrt{10})\), 6.0.23104000.1 |
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: | deg 18, 18.0.37133262473195501387776000000.1 |
| 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.4.24b1.2 | $x^{12} + 4 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} + 12 x^{6} + 12 x^{5} + 17 x^{4} + 16 x^{3} + 8 x^{2} + 12 x + 17$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $$[2, 3]^{3}$$ |
|
\(5\)
| 5.3.2.3a1.1 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 23 x + 9$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 5.3.2.3a1.1 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 23 x + 9$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(19\)
| 19.6.1.0a1.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |
| 19.2.3.4a1.3 | $x^{6} + 54 x^{5} + 978 x^{4} + 6048 x^{3} + 1956 x^{2} + 235 x + 331$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
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 | 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *36 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.76.6t1.a.a | $1$ | $ 2^{2} \cdot 19 $ | 6.0.8340544.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.760.6t1.a.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.6.8340544000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.760.6t1.a.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.6.8340544000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.760.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.76.6t1.a.b | $1$ | $ 2^{2} \cdot 19 $ | 6.0.8340544.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.760.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.14440.3t2.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19^{2}$ | 3.1.14440.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.57760.6t3.c.a | $2$ | $ 2^{5} \cdot 5 \cdot 19^{2}$ | 6.2.33362176000.17 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *36 | 2.3040.12t18.b.a | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 12.0.34162868224000000.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.760.6t5.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| *36 | 2.3040.12t18.b.b | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 12.0.34162868224000000.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| *36 | 2.760.6t5.a.b | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (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 *.