Normalized defining polynomial
\( x^{16} - 2 x^{15} - 3 x^{14} + 10 x^{13} + 3 x^{12} - 28 x^{11} + 30 x^{10} + 7 x^{9} - 5 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(780705885094140625\) \(\medspace = 5^{8}\cdot 29^{4}\cdot 41^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(13.13\) | 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))
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Galois root discriminant: | $5^{1/2}29^{1/2}41^{1/2}\approx 77.10382610480494$ | ||
Ramified primes: | \(5\), \(29\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{12}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{3}{8}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{4}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}+\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}+\frac{1}{4}a^{7}+\frac{3}{8}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}$, $\frac{1}{8}a^{14}+\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{3}{8}$, $\frac{1}{58170299704}a^{15}-\frac{2152939319}{58170299704}a^{14}-\frac{27231205}{2644104532}a^{13}+\frac{85542247}{58170299704}a^{12}+\frac{11101350649}{58170299704}a^{11}-\frac{6913687087}{58170299704}a^{10}+\frac{13005553601}{58170299704}a^{9}+\frac{92502693}{2644104532}a^{8}-\frac{9223820185}{29085149852}a^{7}-\frac{5286095905}{29085149852}a^{6}+\frac{17210347951}{58170299704}a^{5}-\frac{2775162713}{29085149852}a^{4}+\frac{298703999}{1322052266}a^{3}+\frac{934544257}{58170299704}a^{2}-\frac{10894003643}{58170299704}a+\frac{18590576797}{58170299704}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{12384977415}{58170299704}a^{15}-\frac{2535945294}{7271287463}a^{14}-\frac{4028529027}{5288209064}a^{13}+\frac{53535312805}{29085149852}a^{12}+\frac{37829421987}{29085149852}a^{11}-\frac{314154652001}{58170299704}a^{10}+\frac{127065394957}{29085149852}a^{9}+\frac{14876598587}{5288209064}a^{8}+\frac{20172768983}{58170299704}a^{7}-\frac{320750268613}{29085149852}a^{6}+\frac{3268761913}{14542574926}a^{5}+\frac{117680860641}{7271287463}a^{4}+\frac{7611801291}{661026133}a^{3}-\frac{15897034741}{58170299704}a^{2}+\frac{25266587494}{7271287463}a-\frac{7957473533}{14542574926}$, $\frac{12384738609}{58170299704}a^{15}-\frac{30925770505}{58170299704}a^{14}-\frac{611340269}{1322052266}a^{13}+\frac{147742996873}{58170299704}a^{12}-\frac{20229693117}{58170299704}a^{11}-\frac{389982952215}{58170299704}a^{10}+\frac{547576981607}{58170299704}a^{9}-\frac{756172733}{1322052266}a^{8}-\frac{24349481532}{7271287463}a^{7}-\frac{308462880761}{29085149852}a^{6}+\frac{632051832065}{58170299704}a^{5}+\frac{454734468107}{29085149852}a^{4}-\frac{6245380705}{1322052266}a^{3}-\frac{602021054527}{58170299704}a^{2}+\frac{488833219275}{58170299704}a-\frac{110365806253}{58170299704}$, $\frac{14922413173}{58170299704}a^{15}-\frac{14660834101}{29085149852}a^{14}-\frac{4271032315}{5288209064}a^{13}+\frac{149179857137}{58170299704}a^{12}+\frac{54897795545}{58170299704}a^{11}-\frac{425617055977}{58170299704}a^{10}+\frac{52885418538}{7271287463}a^{9}+\frac{13818052345}{5288209064}a^{8}-\frac{43842429849}{29085149852}a^{7}-\frac{827551925529}{58170299704}a^{6}+\frac{163715885331}{29085149852}a^{5}+\frac{1179825440301}{58170299704}a^{4}+\frac{36362958761}{5288209064}a^{3}-\frac{50859986465}{7271287463}a^{2}+\frac{336206703257}{58170299704}a-\frac{17945072379}{58170299704}$, $\frac{119403}{29085149852}a^{15}+\frac{10638208153}{58170299704}a^{14}-\frac{1583167951}{5288209064}a^{13}-\frac{40672371263}{58170299704}a^{12}+\frac{95888537091}{58170299704}a^{11}+\frac{37914150107}{29085149852}a^{10}-\frac{293446191693}{58170299704}a^{9}+\frac{17901289519}{5288209064}a^{8}+\frac{214968621239}{58170299704}a^{7}-\frac{3071846963}{7271287463}a^{6}-\frac{618976784413}{58170299704}a^{5}+\frac{15988974457}{29085149852}a^{4}+\frac{21468983287}{1322052266}a^{3}+\frac{293062009893}{29085149852}a^{2}-\frac{286700519323}{58170299704}a+\frac{78535912121}{58170299704}$, $\frac{6433764745}{58170299704}a^{15}-\frac{12774494615}{58170299704}a^{14}-\frac{1782879723}{5288209064}a^{13}+\frac{63868585967}{58170299704}a^{12}+\frac{10986239893}{29085149852}a^{11}-\frac{179789403573}{58170299704}a^{10}+\frac{181598209019}{58170299704}a^{9}+\frac{4735767465}{5288209064}a^{8}-\frac{3057154357}{14542574926}a^{7}-\frac{370556157591}{58170299704}a^{6}+\frac{16195641992}{7271287463}a^{5}+\frac{65821702759}{7271287463}a^{4}+\frac{3964043801}{1322052266}a^{3}-\frac{180479827255}{58170299704}a^{2}+\frac{47101665913}{58170299704}a-\frac{11349069473}{29085149852}$, $\frac{3648645213}{29085149852}a^{15}-\frac{24954720969}{58170299704}a^{14}-\frac{80682341}{1322052266}a^{13}+\frac{108676470939}{58170299704}a^{12}-\frac{18353981209}{14542574926}a^{11}-\frac{32531733530}{7271287463}a^{10}+\frac{497248713567}{58170299704}a^{9}-\frac{8365699169}{2644104532}a^{8}-\frac{171838075889}{58170299704}a^{7}-\frac{384182266555}{58170299704}a^{6}+\frac{187546792883}{14542574926}a^{5}+\frac{56194104640}{7271287463}a^{4}-\frac{14776602489}{1322052266}a^{3}-\frac{308666535229}{29085149852}a^{2}+\frac{362952650023}{58170299704}a-\frac{84387763181}{29085149852}$, $\frac{8707213541}{29085149852}a^{15}-\frac{3739077895}{7271287463}a^{14}-\frac{5688910733}{5288209064}a^{13}+\frac{160346312789}{58170299704}a^{12}+\frac{25869345147}{14542574926}a^{11}-\frac{239108844971}{29085149852}a^{10}+\frac{190654363015}{29085149852}a^{9}+\frac{26056398953}{5288209064}a^{8}-\frac{63651702605}{58170299704}a^{7}-\frac{480311006455}{29085149852}a^{6}+\frac{155019822107}{58170299704}a^{5}+\frac{1432794609039}{58170299704}a^{4}+\frac{70775054305}{5288209064}a^{3}-\frac{338753310395}{58170299704}a^{2}+\frac{328017502875}{58170299704}a+\frac{34445097361}{29085149852}$ | 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: | \( 159.84129798 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 159.84129798 \cdot 1}{2\cdot\sqrt{780705885094140625}}\cr\approx \mathstrut & 0.21971222501 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 16T388):
A solvable group of order 128 |
The 20 conjugacy class representatives for $C_2\wr D_4$ |
Character table for $C_2\wr D_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.725.1, 4.0.1025.1, 4.0.29725.2, 8.4.21550625.1 x2, 8.0.30468125.1 x2, 8.0.883575625.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 8.4.21550625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(29\) | 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(41\) | 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |