Normalized defining polynomial
\( x^{16} - 3x^{12} - 27x^{8} + 12x^{4} + 16 \)
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)
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Discriminant: | \(11161211905600000000\) \(\medspace = 2^{12}\cdot 5^{8}\cdot 17^{8}\) | 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: | \(15.51\) | 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: | $2\cdot 5^{1/2}17^{1/2}\approx 18.439088914585774$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{296}a^{12}+\frac{59}{296}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{5}{296}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{25}{74}$, $\frac{1}{592}a^{13}-\frac{1}{8}a^{11}+\frac{133}{592}a^{9}-\frac{1}{8}a^{7}+\frac{5}{592}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{49}{148}a-\frac{1}{2}$, $\frac{1}{592}a^{14}-\frac{15}{592}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{5}{592}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{3}{37}a^{2}-\frac{1}{2}$, $\frac{1}{1184}a^{15}-\frac{15}{1184}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{143}{1184}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{31}{148}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$
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: | $\frac{1}{148}a^{12}-\frac{15}{148}a^{8}+\frac{5}{148}a^{4}+\frac{25}{37}$, $\frac{1}{296}a^{14}-\frac{3}{74}a^{12}-\frac{15}{296}a^{10}+\frac{4}{37}a^{8}+\frac{5}{296}a^{6}+\frac{48}{37}a^{4}+\frac{31}{37}a^{2}-\frac{2}{37}$, $\frac{27}{592}a^{15}-\frac{109}{592}a^{11}-\frac{605}{592}a^{7}+\frac{97}{74}a^{3}+\frac{1}{2}a$, $\frac{27}{1184}a^{15}+\frac{1}{592}a^{14}+\frac{1}{296}a^{13}-\frac{3}{74}a^{12}-\frac{109}{1184}a^{11}-\frac{15}{592}a^{10}-\frac{15}{296}a^{9}+\frac{4}{37}a^{8}-\frac{605}{1184}a^{7}+\frac{5}{592}a^{6}+\frac{5}{296}a^{5}+\frac{155}{148}a^{4}+\frac{97}{148}a^{3}+\frac{25}{148}a^{2}+\frac{87}{148}a-\frac{2}{37}$, $\frac{39}{1184}a^{15}-\frac{1}{296}a^{14}+\frac{1}{592}a^{13}+\frac{3}{148}a^{12}-\frac{141}{1184}a^{11}+\frac{15}{296}a^{10}-\frac{15}{592}a^{9}-\frac{2}{37}a^{8}-\frac{989}{1184}a^{7}-\frac{5}{296}a^{6}+\frac{5}{592}a^{5}-\frac{59}{148}a^{4}+\frac{309}{296}a^{3}-\frac{161}{148}a^{2}-\frac{3}{37}a+\frac{1}{37}$, $\frac{39}{1184}a^{15}+\frac{1}{296}a^{14}+\frac{1}{592}a^{13}-\frac{3}{148}a^{12}-\frac{141}{1184}a^{11}-\frac{15}{296}a^{10}-\frac{15}{592}a^{9}+\frac{2}{37}a^{8}-\frac{989}{1184}a^{7}+\frac{5}{296}a^{6}+\frac{5}{592}a^{5}+\frac{59}{148}a^{4}+\frac{309}{296}a^{3}+\frac{161}{148}a^{2}+\frac{34}{37}a-\frac{1}{37}$, $\frac{35}{1184}a^{15}+\frac{1}{592}a^{14}+\frac{13}{592}a^{13}+\frac{3}{74}a^{12}-\frac{81}{1184}a^{11}-\frac{15}{592}a^{10}-\frac{47}{592}a^{9}-\frac{4}{37}a^{8}-\frac{1009}{1184}a^{7}+\frac{5}{592}a^{6}-\frac{231}{592}a^{5}-\frac{155}{148}a^{4}-\frac{13}{296}a^{3}+\frac{99}{148}a^{2}-\frac{2}{37}a-\frac{33}{74}$, $\frac{23}{1184}a^{15}-\frac{5}{296}a^{13}+\frac{3}{148}a^{12}-\frac{49}{1184}a^{11}+\frac{1}{296}a^{9}-\frac{2}{37}a^{8}-\frac{625}{1184}a^{7}+\frac{123}{296}a^{5}-\frac{59}{148}a^{4}+\frac{5}{74}a^{3}-\frac{1}{4}a^{2}+\frac{83}{148}a+\frac{39}{74}$, $\frac{27}{1184}a^{15}+\frac{1}{592}a^{14}-\frac{1}{296}a^{13}-\frac{3}{74}a^{12}-\frac{109}{1184}a^{11}-\frac{15}{592}a^{10}+\frac{15}{296}a^{9}+\frac{4}{37}a^{8}-\frac{605}{1184}a^{7}+\frac{5}{592}a^{6}-\frac{5}{296}a^{5}+\frac{155}{148}a^{4}+\frac{97}{148}a^{3}+\frac{25}{148}a^{2}-\frac{161}{148}a-\frac{2}{37}$ | 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: | \( 1838.66338876 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 1838.66338876 \cdot 1}{2\cdot\sqrt{11161211905600000000}}\cr\approx \mathstrut & 0.270904206618 \end{aligned}\]
Galois group
A solvable group of order 32 |
The 11 conjugacy class representatives for $D_8:C_2$ |
Character table for $D_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), 4.2.28900.1, 4.2.1156.1, \(\Q(\sqrt{5}, \sqrt{17})\), 8.4.835210000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(5\) | 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(17\) | 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 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
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 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |