Normalized defining polynomial
\( x^{16} - 2 x^{15} + 6 x^{13} - 10 x^{12} + 6 x^{11} + 12 x^{10} - 34 x^{9} + 46 x^{8} - 34 x^{7} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(14012498575360000\) \(\medspace = 2^{28}\cdot 5^{4}\cdot 17^{4}\) | 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: | \(10.21\) | 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: | $2^{7/4}5^{1/2}17^{1/2}\approx 31.01072753763937$ | ||
Ramified primes: | \(2\), \(5\), \(17\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{10}a^{14}+\frac{1}{10}a^{13}+\frac{1}{5}a^{12}+\frac{1}{10}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{1}{2}a^{6}+\frac{3}{10}a^{5}-\frac{2}{5}a^{4}+\frac{1}{10}a^{3}-\frac{3}{10}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{15}+\frac{1}{10}a^{13}-\frac{1}{10}a^{12}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{3}{10}a^{7}-\frac{1}{5}a^{6}+\frac{3}{10}a^{5}-\frac{1}{2}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}$
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: | \( \frac{1}{2} a^{15} + \frac{5}{2} a^{14} - \frac{7}{2} a^{13} - 3 a^{12} + \frac{23}{2} a^{11} - \frac{21}{2} a^{10} + \frac{7}{2} a^{9} + 28 a^{8} - \frac{93}{2} a^{7} + \frac{87}{2} a^{6} - \frac{29}{2} a^{5} - 3 a^{4} + \frac{21}{2} a^{3} - \frac{15}{2} a^{2} - \frac{3}{2} a + 3 \) (order $4$) | 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: | $\frac{11}{10}a^{15}-3a^{14}-\frac{9}{10}a^{13}+\frac{89}{10}a^{12}-\frac{23}{2}a^{11}+\frac{16}{5}a^{10}+\frac{86}{5}a^{9}-\frac{226}{5}a^{8}+\frac{467}{10}a^{7}-\frac{136}{5}a^{6}-\frac{17}{10}a^{5}+\frac{23}{2}a^{4}-\frac{139}{10}a^{3}+\frac{17}{5}a^{2}+3a-\frac{13}{5}$, $\frac{17}{5}a^{15}-3a^{14}-\frac{61}{10}a^{13}+\frac{161}{10}a^{12}-\frac{23}{2}a^{11}-\frac{26}{5}a^{10}+\frac{453}{10}a^{9}-\frac{314}{5}a^{8}+\frac{254}{5}a^{7}-\frac{34}{5}a^{6}-\frac{133}{10}a^{5}+\frac{39}{2}a^{4}-\frac{91}{10}a^{3}-\frac{7}{5}a^{2}+\frac{9}{2}a-\frac{7}{5}$, $\frac{13}{10}a^{15}-\frac{12}{5}a^{14}-\frac{8}{5}a^{13}+\frac{79}{10}a^{12}-\frac{47}{5}a^{11}+\frac{11}{5}a^{10}+\frac{179}{10}a^{9}-\frac{391}{10}a^{8}+\frac{389}{10}a^{7}-\frac{118}{5}a^{6}+\frac{6}{5}a^{5}+\frac{61}{10}a^{4}-\frac{38}{5}a^{3}+\frac{2}{5}a^{2}+\frac{11}{10}a-\frac{7}{10}$, $\frac{5}{2}a^{14}-2a^{13}-\frac{9}{2}a^{12}+11a^{11}-\frac{15}{2}a^{10}-3a^{9}+31a^{8}-43a^{7}+\frac{73}{2}a^{6}-9a^{5}-\frac{5}{2}a^{4}+10a^{3}-\frac{11}{2}a^{2}+3$, $\frac{37}{10}a^{15}-\frac{71}{10}a^{14}-\frac{29}{10}a^{13}+\frac{241}{10}a^{12}-\frac{153}{5}a^{11}+\frac{44}{5}a^{10}+\frac{263}{5}a^{9}-\frac{1179}{10}a^{8}+\frac{1301}{10}a^{7}-\frac{719}{10}a^{6}+\frac{53}{10}a^{5}+\frac{279}{10}a^{4}-\frac{127}{5}a^{3}+\frac{43}{5}a^{2}+\frac{32}{5}a-\frac{53}{10}$, $\frac{9}{10}a^{15}-a^{14}-\frac{8}{5}a^{13}+\frac{51}{10}a^{12}-4a^{11}-\frac{11}{5}a^{10}+\frac{143}{10}a^{9}-\frac{99}{5}a^{8}+\frac{143}{10}a^{7}+\frac{11}{5}a^{6}-\frac{54}{5}a^{5}+\frac{19}{2}a^{4}-\frac{18}{5}a^{3}-\frac{7}{5}a^{2}+\frac{5}{2}a-\frac{7}{5}$, $\frac{5}{2}a^{15}-\frac{18}{5}a^{14}-\frac{18}{5}a^{13}+\frac{69}{5}a^{12}-\frac{68}{5}a^{11}+\frac{9}{10}a^{10}+\frac{327}{10}a^{9}-\frac{123}{2}a^{8}+\frac{597}{10}a^{7}-30a^{6}+\frac{16}{5}a^{5}+\frac{42}{5}a^{4}-\frac{43}{5}a^{3}+\frac{3}{10}a^{2}+\frac{39}{10}a-\frac{11}{10}$ | 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: | \( 41.4539267385 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 41.4539267385 \cdot 1}{4\cdot\sqrt{14012498575360000}}\cr\approx \mathstrut & 0.212660504756 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 4.0.320.1, 4.0.272.1, 4.0.5440.1, 8.0.6963200.1 x2, 8.0.5918720.1 x2, 8.0.29593600.2 |
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: | 8.0.5918720.1 |
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 | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $8$ | $2$ | $28$ | |||
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |