Normalized defining polynomial
\( x^{16} - 20 x^{12} - 48 x^{11} + 96 x^{10} + 48 x^{9} + 162 x^{8} - 384 x^{5} + 484 x^{4} + 144 x^{3} + \cdots + 25 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(4357047163233901253492736\)
\(\medspace = 2^{66}\cdot 3^{10}\)
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| Root discriminant: | \(34.67\) |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{36}a^{12}-\frac{1}{36}a^{11}-\frac{1}{9}a^{10}+\frac{1}{36}a^{9}-\frac{1}{4}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}-\frac{5}{12}a^{4}-\frac{13}{36}a^{3}-\frac{1}{9}a^{2}+\frac{1}{36}a-\frac{17}{36}$, $\frac{1}{36}a^{13}+\frac{1}{9}a^{11}-\frac{1}{12}a^{10}+\frac{1}{36}a^{9}-\frac{5}{36}a^{8}+\frac{2}{9}a^{7}+\frac{1}{3}a^{6}-\frac{7}{36}a^{5}+\frac{2}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{12}a^{2}-\frac{7}{36}a-\frac{17}{36}$, $\frac{1}{36}a^{14}+\frac{1}{36}a^{11}-\frac{1}{36}a^{10}-\frac{1}{36}a^{8}+\frac{7}{18}a^{7}-\frac{5}{36}a^{6}-\frac{1}{6}a^{5}-\frac{1}{18}a^{4}-\frac{5}{36}a^{3}+\frac{1}{4}a^{2}+\frac{1}{6}a+\frac{5}{36}$, $\frac{1}{23\cdots 00}a^{15}+\frac{122144297585584}{11\cdots 55}a^{14}-\frac{41293546012319}{31\cdots 88}a^{13}+\frac{29611314987487}{23\cdots 91}a^{12}+\frac{978454207737569}{11\cdots 55}a^{11}-\frac{54\cdots 37}{59\cdots 75}a^{10}-\frac{58\cdots 97}{11\cdots 50}a^{9}+\frac{161170967052598}{66\cdots 75}a^{8}-\frac{88\cdots 23}{23\cdots 00}a^{7}-\frac{455873499455222}{11\cdots 55}a^{6}-\frac{21\cdots 57}{95\cdots 64}a^{5}-\frac{39\cdots 07}{19\cdots 25}a^{4}+\frac{80\cdots 91}{59\cdots 75}a^{3}-\frac{500676310327376}{66\cdots 75}a^{2}+\frac{25\cdots 13}{11\cdots 50}a-\frac{41\cdots 49}{11\cdots 55}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{215303366412091}{11\cdots 50}a^{15}+\frac{95416395822919}{23\cdots 10}a^{14}+\frac{3725883711271}{23\cdots 91}a^{13}-\frac{2208143706023}{47\cdots 82}a^{12}+\frac{588346737154741}{15\cdots 40}a^{11}+\frac{20\cdots 11}{23\cdots 00}a^{10}-\frac{95\cdots 67}{23\cdots 00}a^{9}+\frac{54\cdots 29}{23\cdots 00}a^{8}+\frac{632865938108427}{13\cdots 50}a^{7}+\frac{13\cdots 93}{23\cdots 10}a^{6}+\frac{569715622342730}{23\cdots 91}a^{5}+\frac{11\cdots 19}{11\cdots 50}a^{4}-\frac{53\cdots 73}{23\cdots 00}a^{3}+\frac{21\cdots 27}{23\cdots 00}a^{2}-\frac{29\cdots 07}{23\cdots 00}a+\frac{91\cdots 37}{47\cdots 20}$, $\frac{15\cdots 67}{11\cdots 50}a^{15}-\frac{74443093520569}{11\cdots 55}a^{14}+\frac{151301684587}{264232775590999}a^{13}-\frac{151320281786}{792698326772997}a^{12}-\frac{12\cdots 91}{47\cdots 20}a^{11}-\frac{12\cdots 07}{23\cdots 00}a^{10}+\frac{37\cdots 79}{23\cdots 00}a^{9}+\frac{12\cdots 59}{79\cdots 00}a^{8}+\frac{22\cdots 09}{11\cdots 50}a^{7}-\frac{12\cdots 33}{11\cdots 55}a^{6}+\frac{30659627419816}{23\cdots 91}a^{5}-\frac{10\cdots 88}{19\cdots 25}a^{4}+\frac{69\cdots 67}{79\cdots 00}a^{3}-\frac{29\cdots 99}{23\cdots 00}a^{2}+\frac{33\cdots 59}{23\cdots 00}a-\frac{14\cdots 31}{52\cdots 80}$, $\frac{290453326804667}{11\cdots 50}a^{15}-\frac{16290613950953}{15\cdots 40}a^{14}-\frac{412143557577}{10\cdots 96}a^{13}-\frac{2579656145929}{47\cdots 82}a^{12}+\frac{565152772930414}{11\cdots 55}a^{11}+\frac{33\cdots 57}{23\cdots 00}a^{10}-\frac{44\cdots 81}{26\cdots 00}a^{9}-\frac{19\cdots 51}{11\cdots 50}a^{8}-\frac{48\cdots 09}{11\cdots 50}a^{7}-\frac{81\cdots 23}{47\cdots 20}a^{6}-\frac{13\cdots 59}{31\cdots 88}a^{5}+\frac{40\cdots 03}{11\cdots 50}a^{4}-\frac{59\cdots 69}{59\cdots 75}a^{3}+\frac{62\cdots 49}{23\cdots 00}a^{2}-\frac{93\cdots 03}{79\cdots 00}a+\frac{668308040202347}{23\cdots 10}$, $\frac{242008810931707}{19\cdots 25}a^{15}+\frac{131532024855847}{15\cdots 40}a^{14}-\frac{3056122553141}{10\cdots 96}a^{13}+\frac{1715169326798}{23\cdots 91}a^{12}+\frac{58\cdots 43}{23\cdots 10}a^{11}+\frac{99\cdots 07}{23\cdots 00}a^{10}-\frac{35\cdots 29}{23\cdots 00}a^{9}+\frac{13\cdots 83}{39\cdots 50}a^{8}-\frac{11\cdots 92}{59\cdots 75}a^{7}+\frac{60\cdots 57}{47\cdots 20}a^{6}-\frac{33\cdots 93}{95\cdots 64}a^{5}+\frac{31\cdots 96}{66\cdots 75}a^{4}-\frac{10\cdots 63}{11\cdots 50}a^{3}+\frac{70\cdots 99}{23\cdots 00}a^{2}-\frac{30\cdots 09}{23\cdots 00}a+\frac{11\cdots 87}{23\cdots 10}$, $\frac{16\cdots 67}{11\cdots 50}a^{15}-\frac{224358287423671}{47\cdots 20}a^{14}+\frac{12836361635185}{95\cdots 64}a^{13}-\frac{2616930726904}{23\cdots 91}a^{12}-\frac{362963290164371}{13\cdots 95}a^{11}-\frac{13\cdots 07}{23\cdots 00}a^{10}+\frac{36\cdots 29}{23\cdots 00}a^{9}+\frac{21\cdots 01}{11\cdots 50}a^{8}+\frac{85\cdots 03}{39\cdots 50}a^{7}-\frac{40\cdots 07}{47\cdots 20}a^{6}+\frac{43\cdots 65}{95\cdots 64}a^{5}-\frac{33\cdots 14}{59\cdots 75}a^{4}+\frac{49\cdots 94}{59\cdots 75}a^{3}-\frac{48\cdots 99}{23\cdots 00}a^{2}+\frac{31\cdots 09}{23\cdots 00}a-\frac{10\cdots 97}{23\cdots 10}$, $\frac{227775009946927}{59\cdots 75}a^{15}+\frac{14299454780879}{15\cdots 40}a^{14}+\frac{3736827411501}{528465551181998}a^{13}+\frac{12153786614593}{31\cdots 88}a^{12}+\frac{184691719599999}{26\cdots 90}a^{11}+\frac{38\cdots 59}{23\cdots 00}a^{10}-\frac{64\cdots 49}{11\cdots 50}a^{9}-\frac{12\cdots 99}{23\cdots 00}a^{8}+\frac{133942054241319}{66\cdots 75}a^{7}+\frac{57\cdots 09}{47\cdots 20}a^{6}+\frac{35\cdots 33}{47\cdots 82}a^{5}+\frac{28\cdots 47}{23\cdots 00}a^{4}-\frac{23\cdots 81}{11\cdots 50}a^{3}-\frac{28\cdots 37}{23\cdots 00}a^{2}-\frac{28\cdots 29}{11\cdots 50}a-\frac{90\cdots 47}{47\cdots 20}$, $\frac{247276854904501}{39\cdots 50}a^{15}-\frac{6019614905701}{11\cdots 55}a^{14}+\frac{14649845354647}{95\cdots 64}a^{13}-\frac{18306345233327}{95\cdots 64}a^{12}-\frac{62\cdots 59}{47\cdots 20}a^{11}-\frac{71\cdots 63}{23\cdots 00}a^{10}+\frac{69\cdots 93}{11\cdots 50}a^{9}+\frac{13\cdots 13}{66\cdots 75}a^{8}+\frac{16\cdots 31}{11\cdots 50}a^{7}+\frac{47\cdots 78}{11\cdots 55}a^{6}+\frac{703217310273295}{95\cdots 64}a^{5}-\frac{26\cdots 93}{79\cdots 00}a^{4}+\frac{37\cdots 09}{23\cdots 00}a^{3}-\frac{77\cdots 91}{23\cdots 00}a^{2}+\frac{17\cdots 01}{39\cdots 50}a+\frac{41\cdots 71}{11\cdots 55}$
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| Regulator: | \( 653141.1454777224 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 653141.1454777224 \cdot 2}{2\cdot\sqrt{4357047163233901253492736}}\cr\approx \mathstrut & 0.760063469745645 \end{aligned}\] (assuming GRH)
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
| \(\Q(\sqrt{-2}) \), 4.0.6144.1, 8.0.260919263232.15, 8.0.28991029248.2, 8.0.21743271936.4 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.30 | $x^{16} + 16 x^{12} + 8 x^{10} + 16 x^{5} + 16 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 16T969 | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$ |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |