Normalized defining polynomial
\( x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256 \)
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: | \(45212176000000000000\) \(\medspace = 2^{16}\cdot 5^{12}\cdot 41^{4}\) | 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: | \(16.92\) | 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: | $2\cdot 5^{3/4}41^{1/2}\approx 42.82027255342763$ | ||
Ramified primes: | \(2\), \(5\), \(41\) | 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)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{12}a^{10}+\frac{1}{12}a^{8}-\frac{1}{6}a^{6}+\frac{1}{3}a^{4}+\frac{1}{12}a^{2}$, $\frac{1}{24}a^{11}+\frac{1}{24}a^{9}+\frac{5}{12}a^{7}+\frac{1}{6}a^{5}+\frac{1}{24}a^{3}-\frac{1}{2}a$, $\frac{1}{48}a^{12}+\frac{1}{48}a^{10}+\frac{1}{24}a^{8}+\frac{5}{12}a^{6}+\frac{17}{48}a^{4}-\frac{5}{12}a^{2}+\frac{1}{3}$, $\frac{1}{96}a^{13}+\frac{1}{96}a^{11}+\frac{1}{48}a^{9}-\frac{7}{24}a^{7}-\frac{31}{96}a^{5}-\frac{5}{24}a^{3}+\frac{1}{6}a$, $\frac{1}{192}a^{14}+\frac{1}{192}a^{12}+\frac{1}{96}a^{10}-\frac{7}{48}a^{8}-\frac{31}{192}a^{6}-\frac{5}{48}a^{4}-\frac{5}{12}a^{2}$, $\frac{1}{384}a^{15}+\frac{1}{384}a^{13}+\frac{1}{192}a^{11}-\frac{7}{96}a^{9}+\frac{161}{384}a^{7}+\frac{43}{96}a^{5}-\frac{5}{24}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{48} a^{13} + \frac{1}{16} a^{11} + \frac{1}{12} a^{9} - \frac{1}{6} a^{7} - \frac{23}{48} a^{5} - \frac{3}{8} a^{3} - \frac{1}{6} a \) (order $20$) | 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: | $\frac{5}{192}a^{14}+\frac{3}{64}a^{12}-\frac{3}{32}a^{10}-\frac{25}{48}a^{8}-\frac{139}{192}a^{6}+\frac{1}{2}a^{4}+\frac{8}{3}a^{2}+\frac{11}{3}$, $\frac{1}{384}a^{15}-\frac{5}{192}a^{14}+\frac{1}{384}a^{13}-\frac{3}{64}a^{12}-\frac{7}{192}a^{11}+\frac{3}{32}a^{10}-\frac{11}{96}a^{9}+\frac{25}{48}a^{8}+\frac{1}{384}a^{7}+\frac{139}{192}a^{6}+\frac{9}{32}a^{5}-\frac{1}{2}a^{4}+\frac{3}{4}a^{3}-\frac{8}{3}a^{2}+\frac{1}{2}a-\frac{8}{3}$, $\frac{1}{384}a^{15}+\frac{1}{384}a^{13}-\frac{7}{192}a^{11}-\frac{11}{96}a^{9}+\frac{1}{384}a^{7}+\frac{9}{32}a^{5}+\frac{3}{4}a^{3}+\frac{1}{2}a-1$, $\frac{5}{192}a^{14}+\frac{7}{64}a^{12}+\frac{13}{96}a^{10}-\frac{11}{48}a^{8}-\frac{155}{192}a^{6}-\frac{37}{48}a^{4}-\frac{5}{12}a^{2}-\frac{1}{3}$, $\frac{5}{192}a^{15}+\frac{5}{192}a^{14}+\frac{19}{192}a^{13}+\frac{7}{64}a^{12}+\frac{1}{12}a^{11}+\frac{13}{96}a^{10}-\frac{7}{24}a^{9}-\frac{11}{48}a^{8}-\frac{179}{192}a^{7}-\frac{155}{192}a^{6}-\frac{59}{96}a^{5}-\frac{37}{48}a^{4}+\frac{3}{4}a^{3}-\frac{5}{12}a^{2}+a-\frac{1}{3}$, $\frac{11}{384}a^{15}-\frac{1}{24}a^{14}+\frac{13}{128}a^{13}-\frac{1}{6}a^{12}+\frac{3}{64}a^{11}-\frac{1}{24}a^{10}-\frac{13}{32}a^{9}+\frac{13}{12}a^{8}-\frac{119}{128}a^{7}+\frac{59}{24}a^{6}-\frac{1}{3}a^{5}+\frac{3}{8}a^{4}+\frac{3}{2}a^{3}-6a^{2}+\frac{3}{2}a-9$, $\frac{1}{384}a^{15}+\frac{3}{32}a^{14}-\frac{7}{384}a^{13}+\frac{9}{32}a^{12}-\frac{1}{64}a^{11}-\frac{1}{8}a^{10}+\frac{5}{96}a^{9}-\frac{25}{12}a^{8}+\frac{65}{384}a^{7}-\frac{335}{96}a^{6}-\frac{23}{96}a^{5}+\frac{47}{48}a^{4}-\frac{1}{8}a^{3}+\frac{119}{12}a^{2}-\frac{1}{6}a+\frac{35}{3}$ | 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: | \( 11629.45174747923 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11629.45174747923 \cdot 2}{20\cdot\sqrt{45212176000000000000}}\cr\approx \mathstrut & 0.420117220153158 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 16T21):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$ |
Character table for $C_2 \times (C_2^2:C_4)$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 siblings: | deg 16, deg 16, deg 16 |
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 | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(41\) | 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |