Normalized defining polynomial
\( x^{16} - 4 x^{15} + 11 x^{14} - 24 x^{13} + 46 x^{12} - 75 x^{11} + 104 x^{10} - 121 x^{9} + 121 x^{8} + \cdots + 1 \)
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: | \(5605632412826517\) \(\medspace = 3^{10}\cdot 37^{7}\) | 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: | \(9.64\) | 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: | $3^{3/4}37^{3/4}\approx 34.197332740531884$ | ||
Ramified primes: | \(3\), \(37\) | 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: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}+\frac{2}{13}a^{13}+\frac{6}{13}a^{12}+\frac{4}{13}a^{11}-\frac{6}{13}a^{10}+\frac{3}{13}a^{9}+\frac{3}{13}a^{8}+\frac{2}{13}a^{7}+\frac{4}{13}a^{6}+\frac{5}{13}a^{4}+\frac{6}{13}a^{3}-\frac{6}{13}a^{2}+\frac{4}{13}a-\frac{3}{13}$, $\frac{1}{13}a^{15}+\frac{2}{13}a^{13}+\frac{5}{13}a^{12}-\frac{1}{13}a^{11}+\frac{2}{13}a^{10}-\frac{3}{13}a^{9}-\frac{4}{13}a^{8}+\frac{5}{13}a^{6}+\frac{5}{13}a^{5}-\frac{4}{13}a^{4}-\frac{5}{13}a^{3}+\frac{3}{13}a^{2}+\frac{2}{13}a+\frac{6}{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( \frac{14}{13} a^{15} - \frac{56}{13} a^{14} + \frac{150}{13} a^{13} - \frac{331}{13} a^{12} + \frac{620}{13} a^{11} - 78 a^{10} + \frac{1376}{13} a^{9} - \frac{1589}{13} a^{8} + \frac{1539}{13} a^{7} - \frac{1337}{13} a^{6} + \frac{1032}{13} a^{5} - \frac{752}{13} a^{4} + \frac{491}{13} a^{3} - \frac{298}{13} a^{2} + \frac{142}{13} a - \frac{34}{13} \) (order $6$) | 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: | $\frac{4}{13}a^{14}-\frac{5}{13}a^{13}+\frac{24}{13}a^{12}-\frac{36}{13}a^{11}+\frac{80}{13}a^{10}-\frac{105}{13}a^{9}+\frac{155}{13}a^{8}-\frac{161}{13}a^{7}+\frac{172}{13}a^{6}-10a^{5}+\frac{111}{13}a^{4}-\frac{80}{13}a^{3}+\frac{54}{13}a^{2}-\frac{23}{13}a+\frac{14}{13}$, $\frac{48}{13}a^{15}-\frac{153}{13}a^{14}+\frac{388}{13}a^{13}-\frac{795}{13}a^{12}+\frac{1459}{13}a^{11}-170a^{10}+\frac{2829}{13}a^{9}-\frac{2978}{13}a^{8}+\frac{2736}{13}a^{7}-\frac{2257}{13}a^{6}+\frac{1696}{13}a^{5}-\frac{1152}{13}a^{4}+\frac{766}{13}a^{3}-\frac{446}{13}a^{2}+\frac{160}{13}a-\frac{33}{13}$, $\frac{8}{13}a^{15}-\frac{8}{13}a^{14}+a^{13}+\frac{5}{13}a^{12}-\frac{27}{13}a^{11}+\frac{129}{13}a^{10}-\frac{269}{13}a^{9}+\frac{438}{13}a^{8}-\frac{497}{13}a^{7}+\frac{463}{13}a^{6}-\frac{376}{13}a^{5}+\frac{292}{13}a^{4}-\frac{205}{13}a^{3}+\frac{137}{13}a^{2}-\frac{94}{13}a+\frac{33}{13}$, $\frac{9}{13}a^{15}-\frac{36}{13}a^{14}+\frac{89}{13}a^{13}-\frac{184}{13}a^{12}+\frac{341}{13}a^{11}-40a^{10}+\frac{671}{13}a^{9}-\frac{690}{13}a^{8}+\frac{617}{13}a^{7}-\frac{489}{13}a^{6}+\frac{383}{13}a^{5}-\frac{255}{13}a^{4}+\frac{168}{13}a^{3}-\frac{82}{13}a^{2}+\frac{17}{13}a+\frac{6}{13}$, $\frac{24}{13}a^{15}-\frac{62}{13}a^{14}+\frac{145}{13}a^{13}-\frac{278}{13}a^{12}+\frac{482}{13}a^{11}-\frac{659}{13}a^{10}+\frac{743}{13}a^{9}-\frac{646}{13}a^{8}+\frac{513}{13}a^{7}-\frac{375}{13}a^{6}+\frac{263}{13}a^{5}-\frac{146}{13}a^{4}+\frac{106}{13}a^{3}-\frac{24}{13}a^{2}-\frac{18}{13}a+\frac{18}{13}$, $\frac{24}{13}a^{15}-\frac{56}{13}a^{14}+\frac{144}{13}a^{13}-\frac{268}{13}a^{12}+\frac{480}{13}a^{11}-\frac{656}{13}a^{10}+\frac{774}{13}a^{9}-\frac{706}{13}a^{8}+\frac{590}{13}a^{7}-33a^{6}+\frac{276}{13}a^{5}-\frac{155}{13}a^{4}+\frac{90}{13}a^{3}-\frac{21}{13}a^{2}-\frac{20}{13}a+1$, $\frac{20}{13}a^{15}-\frac{70}{13}a^{14}+\frac{173}{13}a^{13}-\frac{359}{13}a^{12}+\frac{649}{13}a^{11}-\frac{996}{13}a^{10}+\frac{1251}{13}a^{9}-\frac{1304}{13}a^{8}+\frac{1147}{13}a^{7}-\frac{934}{13}a^{6}+\frac{685}{13}a^{5}-\frac{469}{13}a^{4}+23a^{3}-\frac{183}{13}a^{2}+\frac{59}{13}a-\frac{8}{13}$ | 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: | \( 32.1900879299 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 32.1900879299 \cdot 1}{6\cdot\sqrt{5605632412826517}}\cr\approx \mathstrut & 0.174059427860 \end{aligned}\]
Galois group
$C_2^4.D_8$ (as 16T675):
A solvable group of order 256 |
The 31 conjugacy class representatives for $C_2^4.D_8$ |
Character table for $C_2^4.D_8$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.4102893.1 |
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 | $16$ | R | $16$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | $16$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $16$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | $16$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(37\) | 37.4.3.4 | $x^{4} + 185$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
37.8.4.1 | $x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |