Normalized defining polynomial
\( x^{16} - 8 x^{14} - 48 x^{13} - 28 x^{12} + 144 x^{11} + 672 x^{10} + 1488 x^{9} + 2426 x^{8} + \cdots + 67 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{350508149390767}a^{15}-\frac{74392393712947}{350508149390767}a^{14}+\frac{96222025946419}{350508149390767}a^{13}-\frac{102916419408159}{350508149390767}a^{12}-\frac{80765743544398}{350508149390767}a^{11}+\frac{35352911127421}{350508149390767}a^{10}-\frac{105316998545333}{350508149390767}a^{9}-\frac{46525038493351}{350508149390767}a^{8}-\frac{15980335899125}{350508149390767}a^{7}-\frac{174711857628807}{350508149390767}a^{6}+\frac{150770918436124}{350508149390767}a^{5}-\frac{53145444271048}{350508149390767}a^{4}-\frac{120664877848028}{350508149390767}a^{3}+\frac{109927043553416}{350508149390767}a^{2}+\frac{9724016742593}{350508149390767}a+\frac{112271686054231}{350508149390767}$
| 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: |
$\frac{15785570267944}{20618126434751}a^{15}-\frac{8600653228608}{20618126434751}a^{14}+\frac{142535632585904}{20618126434751}a^{13}+\frac{830616746632146}{20618126434751}a^{12}+\frac{708993577778440}{20618126434751}a^{11}-\frac{28\cdots 47}{20618126434751}a^{10}-\frac{12\cdots 40}{20618126434751}a^{9}-\frac{26\cdots 25}{20618126434751}a^{8}-\frac{39\cdots 16}{20618126434751}a^{7}-\frac{45\cdots 78}{20618126434751}a^{6}-\frac{42\cdots 72}{20618126434751}a^{5}-\frac{31\cdots 42}{20618126434751}a^{4}-\frac{19\cdots 34}{20618126434751}a^{3}-\frac{94\cdots 45}{20618126434751}a^{2}-\frac{35\cdots 98}{20618126434751}a-\frac{914724827581960}{20618126434751}$, $\frac{94116745283451}{350508149390767}a^{15}-\frac{48499884989045}{350508149390767}a^{14}-\frac{848449198815327}{350508149390767}a^{13}-\frac{40\cdots 39}{350508149390767}a^{12}+\frac{517373550184476}{350508149390767}a^{11}+\frac{18\cdots 34}{350508149390767}a^{10}+\frac{54\cdots 35}{350508149390767}a^{9}+\frac{89\cdots 11}{350508149390767}a^{8}+\frac{10\cdots 35}{350508149390767}a^{7}+\frac{98\cdots 18}{350508149390767}a^{6}+\frac{71\cdots 60}{350508149390767}a^{5}+\frac{37\cdots 13}{350508149390767}a^{4}+\frac{16\cdots 96}{350508149390767}a^{3}-\frac{282106701620633}{350508149390767}a^{2}+\frac{583255333575984}{350508149390767}a-\frac{23\cdots 56}{350508149390767}$, $\frac{237006041725784}{350508149390767}a^{15}-\frac{11049729705630}{350508149390767}a^{14}-\frac{21\cdots 89}{350508149390767}a^{13}-\frac{11\cdots 41}{350508149390767}a^{12}-\frac{40\cdots 31}{350508149390767}a^{11}+\frac{44\cdots 55}{350508149390767}a^{10}+\frac{15\cdots 44}{350508149390767}a^{9}+\frac{30\cdots 23}{350508149390767}a^{8}+\frac{41\cdots 40}{350508149390767}a^{7}+\frac{44\cdots 85}{350508149390767}a^{6}+\frac{39\cdots 56}{350508149390767}a^{5}+\frac{28\cdots 81}{350508149390767}a^{4}+\frac{16\cdots 19}{350508149390767}a^{3}+\frac{75\cdots 50}{350508149390767}a^{2}+\frac{28\cdots 43}{350508149390767}a+\frac{50\cdots 06}{350508149390767}$, $\frac{152659104119288}{350508149390767}a^{15}+\frac{54388134267203}{350508149390767}a^{14}-\frac{13\cdots 45}{350508149390767}a^{13}-\frac{77\cdots 10}{350508149390767}a^{12}-\frac{52\cdots 59}{350508149390767}a^{11}+\frac{28\cdots 80}{350508149390767}a^{10}+\frac{11\cdots 00}{350508149390767}a^{9}+\frac{23\cdots 10}{350508149390767}a^{8}+\frac{33\cdots 18}{350508149390767}a^{7}+\frac{38\cdots 58}{350508149390767}a^{6}+\frac{35\cdots 50}{350508149390767}a^{5}+\frac{26\cdots 56}{350508149390767}a^{4}+\frac{16\cdots 61}{350508149390767}a^{3}+\frac{81\cdots 63}{350508149390767}a^{2}+\frac{30\cdots 17}{350508149390767}a+\frac{84\cdots 59}{350508149390767}$, $\frac{58579355738226}{350508149390767}a^{15}-\frac{85002721640271}{350508149390767}a^{14}-\frac{460496697995946}{350508149390767}a^{13}-\frac{20\cdots 07}{350508149390767}a^{12}+\frac{23\cdots 90}{350508149390767}a^{11}+\frac{98\cdots 64}{350508149390767}a^{10}+\frac{24\cdots 70}{350508149390767}a^{9}+\frac{31\cdots 73}{350508149390767}a^{8}+\frac{33\cdots 15}{350508149390767}a^{7}+\frac{20\cdots 28}{350508149390767}a^{6}+\frac{506211366963168}{350508149390767}a^{5}-\frac{19\cdots 43}{350508149390767}a^{4}-\frac{28\cdots 05}{350508149390767}a^{3}-\frac{32\cdots 43}{350508149390767}a^{2}-\frac{16\cdots 18}{350508149390767}a-\frac{10\cdots 84}{350508149390767}$, $\frac{11981713719641}{350508149390767}a^{15}-\frac{793708019889078}{350508149390767}a^{14}+\frac{201847690868187}{350508149390767}a^{13}+\frac{76\cdots 48}{350508149390767}a^{12}+\frac{37\cdots 63}{350508149390767}a^{11}+\frac{92\cdots 68}{350508149390767}a^{10}-\frac{15\cdots 43}{350508149390767}a^{9}-\frac{54\cdots 25}{350508149390767}a^{8}-\frac{10\cdots 52}{350508149390767}a^{7}-\frac{13\cdots 01}{350508149390767}a^{6}-\frac{14\cdots 09}{350508149390767}a^{5}-\frac{11\cdots 52}{350508149390767}a^{4}-\frac{79\cdots 46}{350508149390767}a^{3}-\frac{44\cdots 27}{350508149390767}a^{2}-\frac{16\cdots 69}{350508149390767}a-\frac{60\cdots 08}{350508149390767}$, $\frac{21734801498668}{350508149390767}a^{15}+\frac{171367977077118}{350508149390767}a^{14}-\frac{244375000739501}{350508149390767}a^{13}-\frac{25\cdots 19}{350508149390767}a^{12}-\frac{81\cdots 97}{350508149390767}a^{11}+\frac{29\cdots 62}{350508149390767}a^{10}+\frac{47\cdots 12}{350508149390767}a^{9}+\frac{13\cdots 02}{350508149390767}a^{8}+\frac{23\cdots 74}{350508149390767}a^{7}+\frac{29\cdots 43}{350508149390767}a^{6}+\frac{29\cdots 10}{350508149390767}a^{5}+\frac{24\cdots 45}{350508149390767}a^{4}+\frac{16\cdots 13}{350508149390767}a^{3}+\frac{91\cdots 89}{350508149390767}a^{2}+\frac{33\cdots 57}{350508149390767}a+\frac{13\cdots 83}{350508149390767}$
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| Regulator: | \( 435835.4858500988 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 435835.4858500988 \cdot 2}{2\cdot\sqrt{4357047163233901253492736}}\cr\approx \mathstrut & 0.507183836001041 \end{aligned}\]
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
| \(\Q(\sqrt{-2}) \), 4.0.6144.1, 8.0.260919263232.15, 8.0.28991029248.2, 8.0.21743271936.7 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.16.66j1.143 | $x^{16} + 8 x^{14} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | 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 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |