Normalized defining polynomial
\( x^{16} + 8 x^{14} - 16 x^{13} + 24 x^{12} - 16 x^{11} + 32 x^{10} - 224 x^{9} + 336 x^{8} - 416 x^{7} + \cdots + 24 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(1089261790808475313373184\)
\(\medspace = 2^{64}\cdot 3^{10}\)
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| Root discriminant: | \(31.79\) |
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| Galois root discriminant: | $2^{555/128}3^{3/4}\approx 46.035004279893066$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{4}a^{13}$, $\frac{1}{4}a^{14}$, $\frac{1}{482526573790652}a^{15}+\frac{13860706728765}{120631643447663}a^{14}+\frac{335771516965}{120631643447663}a^{13}+\frac{3812024069629}{482526573790652}a^{12}-\frac{17448298629259}{241263286895326}a^{11}+\frac{48933275903869}{241263286895326}a^{10}+\frac{17348994376087}{241263286895326}a^{9}+\frac{10960075807179}{241263286895326}a^{8}+\frac{56898284772889}{241263286895326}a^{7}+\frac{27230576130457}{120631643447663}a^{6}+\frac{50238382660565}{120631643447663}a^{5}-\frac{28258040802123}{120631643447663}a^{4}+\frac{46689473246551}{120631643447663}a^{3}-\frac{16685332740264}{120631643447663}a^{2}+\frac{27510255253797}{120631643447663}a-\frac{44981679796171}{120631643447663}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
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| Narrow class group: | $C_{2}$, which has order $2$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{97070952371}{120631643447663}a^{15}-\frac{236753233718}{120631643447663}a^{14}-\frac{1789574943069}{241263286895326}a^{13}-\frac{551497178944}{120631643447663}a^{12}+\frac{391673963917}{120631643447663}a^{11}-\frac{3918446676008}{120631643447663}a^{10}+\frac{915605280953}{241263286895326}a^{9}+\frac{12699578014720}{120631643447663}a^{8}+\frac{16996148643674}{120631643447663}a^{7}-\frac{26917204437516}{120631643447663}a^{6}+\frac{41903427747032}{120631643447663}a^{5}-\frac{109042129785708}{120631643447663}a^{4}+\frac{28896224010068}{120631643447663}a^{3}-\frac{200869296364808}{120631643447663}a^{2}+\frac{7334728528258}{120631643447663}a-\frac{78702893172311}{120631643447663}$, $\frac{40798904348953}{241263286895326}a^{15}-\frac{10730059391593}{482526573790652}a^{14}-\frac{622841523333289}{482526573790652}a^{13}+\frac{609839207171383}{241263286895326}a^{12}-\frac{777911367144597}{241263286895326}a^{11}+\frac{283937918849489}{241263286895326}a^{10}-\frac{447444327172162}{120631643447663}a^{9}+\frac{87\cdots 35}{241263286895326}a^{8}-\frac{12\cdots 71}{241263286895326}a^{7}+\frac{59\cdots 33}{120631643447663}a^{6}-\frac{75\cdots 46}{120631643447663}a^{5}+\frac{10\cdots 44}{120631643447663}a^{4}-\frac{40\cdots 51}{120631643447663}a^{3}-\frac{933843331815187}{120631643447663}a^{2}-\frac{11\cdots 60}{120631643447663}a-\frac{492020537886187}{120631643447663}$, $\frac{16392512466262}{120631643447663}a^{15}-\frac{19454648052567}{482526573790652}a^{14}-\frac{513566749153607}{482526573790652}a^{13}+\frac{231572546213717}{120631643447663}a^{12}-\frac{578479657895881}{241263286895326}a^{11}+\frac{325538929723999}{241263286895326}a^{10}-\frac{483007170873522}{120631643447663}a^{9}+\frac{71\cdots 21}{241263286895326}a^{8}-\frac{86\cdots 81}{241263286895326}a^{7}+\frac{47\cdots 64}{120631643447663}a^{6}-\frac{77\cdots 86}{120631643447663}a^{5}+\frac{20\cdots 92}{120631643447663}a^{4}-\frac{48\cdots 05}{120631643447663}a^{3}+\frac{932743566870153}{120631643447663}a^{2}-\frac{10\cdots 60}{120631643447663}a+\frac{402721742623691}{120631643447663}$, $\frac{418639232507}{120631643447663}a^{15}-\frac{16370825065245}{482526573790652}a^{14}-\frac{5420168081087}{482526573790652}a^{13}-\frac{54968988763867}{241263286895326}a^{12}+\frac{70761435141553}{120631643447663}a^{11}-\frac{264766720515019}{241263286895326}a^{10}+\frac{118281447907403}{120631643447663}a^{9}-\frac{73398831283479}{120631643447663}a^{8}+\frac{16\cdots 17}{241263286895326}a^{7}-\frac{16\cdots 27}{120631643447663}a^{6}+\frac{24\cdots 25}{120631643447663}a^{5}-\frac{32\cdots 95}{120631643447663}a^{4}+\frac{18\cdots 73}{120631643447663}a^{3}-\frac{17\cdots 04}{120631643447663}a^{2}+\frac{305713159931242}{120631643447663}a-\frac{228808149187751}{120631643447663}$, $\frac{3773088430520}{120631643447663}a^{15}-\frac{5116824502092}{120631643447663}a^{14}+\frac{104358568904815}{482526573790652}a^{13}-\frac{204811638857485}{241263286895326}a^{12}+\frac{141966504982536}{120631643447663}a^{11}-\frac{123445685228974}{120631643447663}a^{10}+\frac{139904970437798}{120631643447663}a^{9}-\frac{985285463755255}{120631643447663}a^{8}+\frac{46\cdots 19}{241263286895326}a^{7}-\frac{23\cdots 06}{120631643447663}a^{6}+\frac{31\cdots 10}{120631643447663}a^{5}-\frac{29\cdots 72}{120631643447663}a^{4}+\frac{11\cdots 49}{120631643447663}a^{3}-\frac{18\cdots 20}{120631643447663}a^{2}+\frac{44649397631248}{120631643447663}a-\frac{320871932241199}{120631643447663}$, $\frac{23540203313997}{482526573790652}a^{15}-\frac{5437622562416}{120631643447663}a^{14}-\frac{38424893521639}{120631643447663}a^{13}+\frac{97936851880483}{241263286895326}a^{12}+\frac{11697016899299}{120631643447663}a^{11}-\frac{371823518731211}{241263286895326}a^{10}+\frac{107126693233820}{120631643447663}a^{9}+\frac{10\cdots 38}{120631643447663}a^{8}-\frac{11\cdots 83}{241263286895326}a^{7}-\frac{12\cdots 44}{120631643447663}a^{6}+\frac{22\cdots 29}{120631643447663}a^{5}-\frac{48\cdots 35}{120631643447663}a^{4}+\frac{28\cdots 26}{120631643447663}a^{3}-\frac{27\cdots 05}{120631643447663}a^{2}+\frac{596415429974704}{120631643447663}a-\frac{336732143879435}{120631643447663}$, $\frac{95438994792583}{482526573790652}a^{15}+\frac{23143691733689}{241263286895326}a^{14}+\frac{182134554923675}{120631643447663}a^{13}-\frac{571393558556557}{241263286895326}a^{12}+\frac{321086624159774}{120631643447663}a^{11}+\frac{119090428077491}{241263286895326}a^{10}+\frac{340913719971948}{120631643447663}a^{9}-\frac{48\cdots 10}{120631643447663}a^{8}+\frac{10\cdots 21}{241263286895326}a^{7}-\frac{40\cdots 22}{120631643447663}a^{6}+\frac{45\cdots 17}{120631643447663}a^{5}+\frac{41\cdots 19}{120631643447663}a^{4}+\frac{24\cdots 82}{120631643447663}a^{3}+\frac{44\cdots 89}{120631643447663}a^{2}+\frac{10\cdots 68}{120631643447663}a+\frac{18\cdots 41}{120631643447663}$
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| Regulator: | \( 469407.7244088967 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 469407.7244088967 \cdot 2}{2\cdot\sqrt{1089261790808475313373184}}\cr\approx \mathstrut & 1.09250402063914 \end{aligned}\]
Galois group
$C_2^6:D_4$ (as 16T969):
| A solvable group of order 512 |
| The 44 conjugacy class representatives for $C_2^6:D_4$ |
| Character table for $C_2^6:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.3072.2, 8.0.14495514624.4, 8.0.130459631616.5, 8.0.5435817984.12 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.64l1.992 | $x^{16} + 8 x^{14} + 20 x^{12} + 16 x^{11} + 4 x^{8} + 8 x^{6} + 16 x^{5} + 8 x^{4} + 16 x + 2$ | $16$ | $1$ | $64$ | 16T969 | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
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\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |