Normalized defining polynomial
\( x^{16} - 2 x^{15} + 4 x^{14} - 5 x^{13} - 2 x^{12} - x^{11} - 8 x^{10} + 3 x^{9} + 6 x^{8} + 10 x^{7} + \cdots - 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(499382553081640625\) \(\medspace = 5^{8}\cdot 89\cdot 119851^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.77\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $5^{1/2}89^{1/2}119851^{1/2}\approx 7302.99219498419$ | ||
Ramified primes: | \(5\), \(89\), \(119851\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{89}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{268229993}a^{15}+\frac{114994542}{268229993}a^{14}+\frac{1567033}{4397213}a^{13}-\frac{92361020}{268229993}a^{12}+\frac{65728456}{268229993}a^{11}+\frac{14376041}{268229993}a^{10}-\frac{79197010}{268229993}a^{9}-\frac{101354724}{268229993}a^{8}-\frac{102032734}{268229993}a^{7}-\frac{67824888}{268229993}a^{6}-\frac{112514464}{268229993}a^{5}+\frac{71278165}{268229993}a^{4}+\frac{59318472}{268229993}a^{3}+\frac{14732274}{268229993}a^{2}-\frac{117025207}{268229993}a-\frac{92204831}{268229993}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{20059440}{268229993}a^{15}-\frac{55834871}{268229993}a^{14}+\frac{1917258}{4397213}a^{13}-\frac{178278990}{268229993}a^{12}+\frac{72192916}{268229993}a^{11}-\frac{74747225}{268229993}a^{10}-\frac{13123349}{268229993}a^{9}-\frac{33052880}{268229993}a^{8}+\frac{336057778}{268229993}a^{7}+\frac{42421537}{268229993}a^{6}-\frac{121590505}{268229993}a^{5}+\frac{96441100}{268229993}a^{4}-\frac{546661599}{268229993}a^{3}-\frac{423731211}{268229993}a^{2}-\frac{3695763}{268229993}a+\frac{227506923}{268229993}$, $\frac{53947060}{268229993}a^{15}-\frac{188327627}{268229993}a^{14}+\frac{7598810}{4397213}a^{13}-\frac{812222269}{268229993}a^{12}+\frac{759815594}{268229993}a^{11}-\frac{570951118}{268229993}a^{10}-\frac{198628644}{268229993}a^{9}+\frac{707235604}{268229993}a^{8}-\frac{633529502}{268229993}a^{7}+\frac{732692971}{268229993}a^{6}-\frac{283740086}{268229993}a^{5}+\frac{168244345}{268229993}a^{4}-\frac{679875748}{268229993}a^{3}-\frac{185774623}{268229993}a^{2}-\frac{406826143}{268229993}a-\frac{247178052}{268229993}$, $\frac{21993941}{268229993}a^{15}-\frac{80495865}{268229993}a^{14}+\frac{3372230}{4397213}a^{13}-\frac{373812822}{268229993}a^{12}+\frac{358401645}{268229993}a^{11}-\frac{231190910}{268229993}a^{10}-\frac{173683591}{268229993}a^{9}+\frac{448757522}{268229993}a^{8}-\frac{221409032}{268229993}a^{7}+\frac{112136529}{268229993}a^{6}+\frac{54786769}{268229993}a^{5}+\frac{126560220}{268229993}a^{4}-\frac{644794450}{268229993}a^{3}+\frac{6527806}{268229993}a^{2}-\frac{74260449}{268229993}a-\frac{115452331}{268229993}$, $\frac{33969943}{268229993}a^{15}-\frac{77954639}{268229993}a^{14}+\frac{3104217}{4397213}a^{13}-\frac{286351175}{268229993}a^{12}+\frac{111197191}{268229993}a^{11}-\frac{179879773}{268229993}a^{10}-\frac{322749667}{268229993}a^{9}+\frac{209276897}{268229993}a^{8}+\frac{84802405}{268229993}a^{7}+\frac{349320905}{268229993}a^{6}+\frac{364208809}{268229993}a^{5}-\frac{163376321}{268229993}a^{4}-\frac{522126244}{268229993}a^{3}-\frac{965999228}{268229993}a^{2}-\frac{789047646}{268229993}a-\frac{412415481}{268229993}$, $\frac{14904067}{268229993}a^{15}-\frac{28120381}{268229993}a^{14}+\frac{1172383}{4397213}a^{13}-\frac{115572298}{268229993}a^{12}+\frac{46725735}{268229993}a^{11}-\frac{103689667}{268229993}a^{10}-\frac{193383464}{268229993}a^{9}+\frac{228295410}{268229993}a^{8}-\frac{117874964}{268229993}a^{7}+\frac{165709905}{268229993}a^{6}+\frac{379072228}{268229993}a^{5}-\frac{264989186}{268229993}a^{4}-\frac{112362362}{268229993}a^{3}-\frac{395901491}{268229993}a^{2}-\frac{480753991}{268229993}a-\frac{232360882}{268229993}$, $\frac{722548}{4397213}a^{15}-\frac{933034}{4397213}a^{14}+\frac{1169878}{4397213}a^{13}+\frac{202400}{4397213}a^{12}-\frac{7061435}{4397213}a^{11}+\frac{1702236}{4397213}a^{10}-\frac{4126485}{4397213}a^{9}-\frac{5375472}{4397213}a^{8}+\frac{10944230}{4397213}a^{7}+\frac{8233004}{4397213}a^{6}+\frac{2103953}{4397213}a^{5}+\frac{9285}{4397213}a^{4}-\frac{9685962}{4397213}a^{3}-\frac{24499526}{4397213}a^{2}-\frac{16350824}{4397213}a-\frac{4616886}{4397213}$, $\frac{66069369}{268229993}a^{15}-\frac{137410939}{268229993}a^{14}+\frac{4542108}{4397213}a^{13}-\frac{361466422}{268229993}a^{12}-\frac{72345890}{268229993}a^{11}-\frac{127354514}{268229993}a^{10}-\frac{425399176}{268229993}a^{9}+\frac{120853730}{268229993}a^{8}+\frac{446188998}{268229993}a^{7}+\frac{614349773}{268229993}a^{6}+\frac{183127902}{268229993}a^{5}+\frac{127126605}{268229993}a^{4}-\frac{1235638132}{268229993}a^{3}-\frac{1219713287}{268229993}a^{2}-\frac{1071660713}{268229993}a-\frac{397082377}{268229993}$, $\frac{79294920}{268229993}a^{15}-\frac{163618241}{268229993}a^{14}+\frac{5446099}{4397213}a^{13}-\frac{452706596}{268229993}a^{12}-\frac{75790635}{268229993}a^{11}-\frac{139249071}{268229993}a^{10}-\frac{613726315}{268229993}a^{9}+\frac{465538488}{268229993}a^{8}+\frac{395476243}{268229993}a^{7}+\frac{815606903}{268229993}a^{6}+\frac{370944170}{268229993}a^{5}-\frac{372188001}{268229993}a^{4}-\frac{1378946446}{268229993}a^{3}-\frac{1581955499}{268229993}a^{2}-\frac{1180925701}{268229993}a-\frac{441913505}{268229993}$, $\frac{50049863}{268229993}a^{15}-\frac{45961945}{268229993}a^{14}+\frac{992442}{4397213}a^{13}-\frac{10737756}{268229993}a^{12}-\frac{361857133}{268229993}a^{11}-\frac{295020320}{268229993}a^{10}+\frac{67497065}{268229993}a^{9}-\frac{550517435}{268229993}a^{8}+\frac{795894316}{268229993}a^{7}+\frac{972590043}{268229993}a^{6}+\frac{157380614}{268229993}a^{5}-\frac{748514}{268229993}a^{4}-\frac{1090381717}{268229993}a^{3}-\frac{1739188909}{268229993}a^{2}-\frac{1578279264}{268229993}a-\frac{335831634}{268229993}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 239.201457412 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 239.201457412 \cdot 1}{2\cdot\sqrt{499382553081640625}}\cr\approx \mathstrut & 0.166615858761 \end{aligned}\]
Galois group
$C_2^6.S_4^2:D_4$ (as 16T1905):
A solvable group of order 294912 |
The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$ |
Character table for $C_2^6.S_4^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 8.6.74906875.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | R | $16$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
\(89\) | $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(119851\) | $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |