Normalized defining polynomial
\( x^{12} + 17x^{10} + 102x^{8} + 238x^{6} + 289x^{4} - 255x^{2} - 136 \)
Invariants
| Degree: | $12$ |
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| Signature: | $[2, 5]$ |
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| Discriminant: |
\(-1123021498208518144\)
\(\medspace = -\,2^{15}\cdot 17^{11}\)
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| Root discriminant: | \(31.93\) |
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| Galois root discriminant: | $2^{2}17^{11/12}\approx 53.69965587306223$ | ||
| Ramified primes: |
\(2\), \(17\)
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| Discriminant root field: | \(\Q(\sqrt{-34}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{8}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{8}-\frac{1}{16}a^{6}-\frac{1}{16}a^{4}-\frac{7}{16}a^{2}-\frac{1}{2}$, $\frac{1}{32}a^{9}-\frac{1}{32}a^{8}+\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{1}{32}a^{5}+\frac{1}{32}a^{4}+\frac{7}{32}a^{3}-\frac{7}{32}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{12736}a^{10}-\frac{15}{3184}a^{8}-\frac{27}{6368}a^{6}-\frac{95}{3184}a^{4}-\frac{3883}{12736}a^{2}-\frac{269}{1592}$, $\frac{1}{12736}a^{11}-\frac{15}{3184}a^{9}-\frac{27}{6368}a^{7}-\frac{95}{3184}a^{5}+\frac{2485}{12736}a^{3}+\frac{527}{1592}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
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Unit group
| Rank: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{65}{12736}a^{10}+\frac{219}{3184}a^{8}+\frac{1429}{6368}a^{6}-\frac{1001}{3184}a^{4}-\frac{4043}{12736}a^{2}+\frac{27}{1592}$, $\frac{93}{12736}a^{10}+\frac{99}{796}a^{8}+\frac{5051}{6368}a^{6}+\frac{3443}{1592}a^{4}+\frac{44045}{12736}a^{2}+\frac{1251}{1592}$, $\frac{203}{12736}a^{11}+\frac{59}{6368}a^{10}+\frac{1671}{6368}a^{9}+\frac{1037}{6368}a^{8}+\frac{2361}{1592}a^{7}+\frac{6167}{6368}a^{6}+\frac{18941}{6368}a^{5}+\frac{12803}{6368}a^{4}+\frac{41581}{12736}a^{3}+\frac{4951}{3184}a^{2}-\frac{8041}{1592}a-\frac{1269}{398}$, $\frac{203}{12736}a^{11}-\frac{59}{6368}a^{10}+\frac{1671}{6368}a^{9}-\frac{1037}{6368}a^{8}+\frac{2361}{1592}a^{7}-\frac{6167}{6368}a^{6}+\frac{18941}{6368}a^{5}-\frac{12803}{6368}a^{4}+\frac{41581}{12736}a^{3}-\frac{4951}{3184}a^{2}-\frac{8041}{1592}a+\frac{1269}{398}$, $\frac{19}{3184}a^{11}-\frac{167}{6368}a^{10}+\frac{705}{6368}a^{9}-\frac{2915}{6368}a^{8}+\frac{4913}{6368}a^{7}-\frac{18245}{6368}a^{6}+\frac{16007}{6368}a^{5}-\frac{46189}{6368}a^{4}+\frac{33337}{6368}a^{3}-\frac{14945}{1592}a^{2}+\frac{4305}{796}a+\frac{634}{199}$, $\frac{159}{6368}a^{11}-\frac{45}{6368}a^{10}+\frac{2599}{6368}a^{9}-\frac{683}{6368}a^{8}+\frac{14697}{6368}a^{7}-\frac{3341}{6368}a^{6}+\frac{30921}{6368}a^{5}-\frac{3397}{6368}a^{4}+\frac{5087}{796}a^{3}+\frac{1101}{3184}a^{2}-\frac{3177}{398}a-\frac{415}{398}$
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| Regulator: | \( 57352.8535259 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 57352.8535259 \cdot 2}{2\cdot\sqrt{1123021498208518144}}\cr\approx \mathstrut & 2.11992418089 \end{aligned}\]
Galois group
$D_6\wr C_2$ (as 12T125):
| A solvable group of order 288 |
| The 27 conjugacy class representatives for $D_6\wr C_2$ |
| Character table for $D_6\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.2.157216.1, 6.2.11358856.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.35094421819016192.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.3.2.6a1.1 | $x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$ | $2$ | $3$ | $6$ | $C_6$ | $$[2]^{3}$$ | |
|
\(17\)
| 17.1.12.11a1.3 | $x^{12} + 153$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $$[\ ]_{12}^{2}$$ |