Normalized defining polynomial
\( x^{16} + 14x^{14} + 84x^{12} + 288x^{10} + 639x^{8} + 941x^{6} + 821x^{4} + 362x^{2} + 181 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(94875856000000000000\)
\(\medspace = 2^{16}\cdot 5^{12}\cdot 181^{3}\)
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| Root discriminant: | \(17.72\) |
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| Galois root discriminant: | $2^{15/8}5^{3/4}181^{3/4}\approx 605.2261387957756$ | ||
| Ramified primes: |
\(2\), \(5\), \(181\)
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| Discriminant root field: | \(\Q(\sqrt{181}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\zeta_{5})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{652773}a^{14}+\frac{100238}{652773}a^{12}+\frac{25642}{217591}a^{10}-\frac{2127}{19781}a^{8}-\frac{322706}{652773}a^{6}+\frac{92073}{217591}a^{4}-\frac{37154}{217591}a^{2}+\frac{172405}{652773}$, $\frac{1}{652773}a^{15}+\frac{100238}{652773}a^{13}+\frac{25642}{217591}a^{11}-\frac{2127}{19781}a^{9}-\frac{322706}{652773}a^{7}+\frac{92073}{217591}a^{5}-\frac{37154}{217591}a^{3}+\frac{172405}{652773}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -\frac{1018}{59343} a^{14} - \frac{3962}{19781} a^{12} - \frac{57032}{59343} a^{10} - \frac{152782}{59343} a^{8} - \frac{88431}{19781} a^{6} - \frac{87060}{19781} a^{4} - \frac{18281}{19781} a^{2} - \frac{11258}{59343} \)
(order $10$)
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| Fundamental units: |
$\frac{2653}{652773}a^{14}+\frac{35212}{652773}a^{12}+\frac{201911}{652773}a^{10}+\frac{63086}{59343}a^{8}+\frac{1604704}{652773}a^{6}+\frac{785340}{217591}a^{4}+\frac{2173393}{652773}a^{2}+\frac{1319729}{652773}$, $\frac{7924}{652773}a^{14}+\frac{26254}{217591}a^{12}+\frac{306824}{652773}a^{10}+\frac{18845}{19781}a^{8}+\frac{659861}{652773}a^{6}-\frac{206104}{652773}a^{4}-\frac{1110974}{652773}a^{2}-\frac{111420}{217591}$, $\frac{10399}{652773}a^{14}+\frac{38024}{217591}a^{12}+\frac{524140}{652773}a^{10}+\frac{127922}{59343}a^{8}+\frac{826600}{217591}a^{6}+\frac{2593682}{652773}a^{4}+\frac{512352}{217591}a^{2}+\frac{977710}{652773}$, $\frac{2653}{652773}a^{15}-\frac{1613}{652773}a^{14}+\frac{35212}{652773}a^{13}-\frac{13781}{652773}a^{12}+\frac{201911}{652773}a^{11}-\frac{18256}{217591}a^{10}+\frac{63086}{59343}a^{9}-\frac{13348}{59343}a^{8}+\frac{1604704}{652773}a^{7}-\frac{388076}{652773}a^{6}+\frac{785340}{217591}a^{5}-\frac{785243}{652773}a^{4}+\frac{2173393}{652773}a^{3}-\frac{812324}{652773}a^{2}+\frac{1319729}{652773}a-\frac{75186}{217591}$, $\frac{3554}{652773}a^{15}+\frac{32656}{652773}a^{14}+\frac{49385}{652773}a^{13}+\frac{368306}{652773}a^{12}+\frac{318299}{652773}a^{11}+\frac{1748089}{652773}a^{10}+\frac{36546}{19781}a^{9}+\frac{430300}{59343}a^{8}+\frac{951240}{217591}a^{7}+\frac{2782878}{217591}a^{6}+\frac{4263554}{652773}a^{5}+\frac{2892133}{217591}a^{4}+\frac{3578719}{652773}a^{3}+\frac{3440632}{652773}a^{2}+\frac{208705}{652773}a+\frac{326170}{217591}$, $\frac{6374}{217591}a^{15}-\frac{942}{19781}a^{14}+\frac{69836}{217591}a^{13}-\frac{9483}{19781}a^{12}+\frac{311392}{217591}a^{11}-\frac{118372}{59343}a^{10}+\frac{209317}{59343}a^{9}-\frac{281036}{59343}a^{8}+\frac{3578084}{652773}a^{7}-\frac{451250}{59343}a^{6}+\frac{2666876}{652773}a^{5}-\frac{137491}{19781}a^{4}-\frac{507701}{652773}a^{3}-\frac{159280}{59343}a^{2}+\frac{1095124}{652773}a-\frac{148967}{59343}$, $\frac{4405}{217591}a^{15}-\frac{3995}{59343}a^{14}+\frac{56251}{217591}a^{13}-\frac{43808}{59343}a^{12}+\frac{287434}{217591}a^{11}-\frac{199564}{59343}a^{10}+\frac{219079}{59343}a^{9}-\frac{172239}{19781}a^{8}+\frac{4358039}{652773}a^{7}-\frac{295523}{19781}a^{6}+\frac{4932062}{652773}a^{5}-\frac{882184}{59343}a^{4}+\frac{1861468}{652773}a^{3}-\frac{296116}{59343}a^{2}+\frac{371896}{652773}a-\frac{60455}{19781}$
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| Regulator: | \( 10757.7141325 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10757.7141325 \cdot 1}{10\cdot\sqrt{94875856000000000000}}\cr\approx \mathstrut & 0.268275549266 \end{aligned}\]
Galois group
$C_4\wr C_4$ (as 16T1192):
| A solvable group of order 1024 |
| The 88 conjugacy class representatives for $C_4\wr C_4$ |
| Character table for $C_4\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.2828125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | $16$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.2.16a5.1 | $x^{16} + 2 x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{9} + 3 x^{8} + 4 x^{7} + 5 x^{6} + 6 x^{5} + 5 x^{4} + 6 x^{3} + 2 x^{2} + 2 x + 3$ | $2$ | $8$ | $16$ | 16T306 | $$[2, 2, 2, 2]^{8}$$ |
|
\(5\)
| 5.4.4.12a1.4 | $x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$ | $4$ | $4$ | $12$ | $C_4^2$ | $$[\ ]_{4}^{4}$$ |
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\(181\)
| 181.1.4.3a1.2 | $x^{4} + 362$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 181.4.1.0a1.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 181.4.1.0a1.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 181.4.1.0a1.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |