Normalized defining polynomial
\( x^{16} - 2x^{15} - x^{14} + 6x^{13} - 16x^{11} + 19x^{10} - 13x^{8} + 19x^{6} - 16x^{5} + 6x^{3} - x^{2} - 2x + 1 \)
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: | \(11272581919277056\) \(\medspace = 2^{24}\cdot 7^{4}\cdot 23^{4}\) | 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: | \(10.08\) | 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: | $2^{3/2}7^{1/2}23^{1/2}\approx 35.888716889852724$ | ||
Ramified primes: | \(2\), \(7\), \(23\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{13}-\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}-\frac{3}{7}a^{5}-\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{15}-\frac{3}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{3}{7}a^{4}-\frac{2}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{3}{7}$
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: | \( -1 \) (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: | $\frac{43}{7}a^{15}-\frac{46}{7}a^{14}-\frac{99}{7}a^{13}+\frac{173}{7}a^{12}+\frac{194}{7}a^{11}-\frac{542}{7}a^{10}+\frac{237}{7}a^{9}+\frac{341}{7}a^{8}-\frac{253}{7}a^{7}-\frac{332}{7}a^{6}+\frac{523}{7}a^{5}-\frac{100}{7}a^{4}-\frac{183}{7}a^{3}+\frac{66}{7}a^{2}+\frac{57}{7}a-5$, $3a^{15}-\frac{29}{7}a^{14}-\frac{45}{7}a^{13}+\frac{99}{7}a^{12}+\frac{79}{7}a^{11}-\frac{299}{7}a^{10}+\frac{171}{7}a^{9}+\frac{155}{7}a^{8}-\frac{159}{7}a^{7}-\frac{167}{7}a^{6}+\frac{304}{7}a^{5}-\frac{89}{7}a^{4}-\frac{96}{7}a^{3}+\frac{36}{7}a^{2}+\frac{32}{7}a-\frac{22}{7}$, $\frac{38}{7}a^{15}-\frac{65}{7}a^{14}-\frac{71}{7}a^{13}+\frac{215}{7}a^{12}+\frac{100}{7}a^{11}-\frac{617}{7}a^{10}+\frac{457}{7}a^{9}+\frac{271}{7}a^{8}-\frac{415}{7}a^{7}-\frac{250}{7}a^{6}+\frac{671}{7}a^{5}-\frac{278}{7}a^{4}-\frac{191}{7}a^{3}+\frac{139}{7}a^{2}+\frac{54}{7}a-\frac{53}{7}$, $\frac{59}{7}a^{15}-\frac{80}{7}a^{14}-\frac{123}{7}a^{13}+\frac{286}{7}a^{12}+\frac{214}{7}a^{11}-\frac{853}{7}a^{10}+\frac{509}{7}a^{9}+\frac{475}{7}a^{8}-73a^{7}-\frac{438}{7}a^{6}+\frac{898}{7}a^{5}-\frac{269}{7}a^{4}-43a^{3}+23a^{2}+\frac{93}{7}a-\frac{68}{7}$, $\frac{8}{7}a^{15}-2a^{14}-\frac{17}{7}a^{13}+\frac{50}{7}a^{12}+\frac{25}{7}a^{11}-\frac{144}{7}a^{10}+\frac{92}{7}a^{9}+\frac{92}{7}a^{8}-\frac{114}{7}a^{7}-10a^{6}+23a^{5}-\frac{45}{7}a^{4}-\frac{72}{7}a^{3}+\frac{34}{7}a^{2}+\frac{20}{7}a-\frac{17}{7}$, $\frac{13}{7}a^{15}+\frac{13}{7}a^{14}-7a^{13}-2a^{12}+\frac{145}{7}a^{11}-\frac{15}{7}a^{10}-\frac{215}{7}a^{9}+23a^{8}+\frac{158}{7}a^{7}-\frac{174}{7}a^{6}-\frac{74}{7}a^{5}+\frac{222}{7}a^{4}-\frac{31}{7}a^{3}-\frac{75}{7}a^{2}+\frac{26}{7}a+\frac{30}{7}$, $a^{15}+\frac{5}{7}a^{14}-\frac{27}{7}a^{13}-\frac{5}{7}a^{12}+\frac{74}{7}a^{11}-\frac{17}{7}a^{10}-\frac{106}{7}a^{9}+\frac{79}{7}a^{8}+\frac{74}{7}a^{7}-\frac{89}{7}a^{6}-\frac{43}{7}a^{5}+\frac{109}{7}a^{4}-\frac{17}{7}a^{3}-\frac{40}{7}a^{2}+\frac{1}{7}a+\frac{19}{7}$ | 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: | \( 15.6225426974 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15.6225426974 \cdot 1}{2\cdot\sqrt{11272581919277056}}\cr\approx \mathstrut & 0.178710047999 \end{aligned}\]
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
\(\Q(\sqrt{2}) \), 4.2.1472.2, 4.2.448.1, 4.0.10304.1, 8.2.15167488.1 x2, 8.2.4616192.1 x2, 8.0.106172416.1 |
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: | 8.2.4616192.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |