Normalized defining polynomial
\( x^{16} + 3x^{14} - 7x^{12} + 24x^{10} + 204x^{8} + 96x^{6} - 112x^{4} + 192x^{2} + 256 \)
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: | \(33672290266555416576\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 23^{8}\) | 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: | \(16.61\) | 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: | $2\cdot 3^{1/2}23^{1/2}\approx 16.61324772583615$ | ||
Ramified primes: | \(2\), \(3\), \(23\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{8}-\frac{1}{4}a^{6}-\frac{5}{12}a^{4}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{24}a^{9}-\frac{1}{8}a^{7}+\frac{7}{24}a^{5}-\frac{1}{4}a^{3}+\frac{1}{6}a$, $\frac{1}{48}a^{10}+\frac{1}{48}a^{8}-\frac{5}{48}a^{6}-\frac{1}{2}a^{5}+\frac{11}{24}a^{4}-\frac{1}{2}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{96}a^{11}+\frac{1}{96}a^{9}-\frac{5}{96}a^{7}-\frac{1}{4}a^{6}-\frac{13}{48}a^{5}+\frac{1}{4}a^{4}-\frac{5}{24}a^{3}-\frac{1}{4}a^{2}-\frac{1}{3}a$, $\frac{1}{3168}a^{12}-\frac{1}{96}a^{10}+\frac{1}{96}a^{8}-\frac{1}{4}a^{7}+\frac{23}{132}a^{6}-\frac{1}{4}a^{5}+\frac{5}{12}a^{4}+\frac{1}{4}a^{3}+\frac{1}{12}a^{2}-\frac{1}{2}a+\frac{46}{99}$, $\frac{1}{6336}a^{13}-\frac{1}{192}a^{11}+\frac{1}{192}a^{9}-\frac{1}{24}a^{8}-\frac{43}{264}a^{7}+\frac{1}{8}a^{6}+\frac{11}{24}a^{5}+\frac{5}{24}a^{4}-\frac{5}{24}a^{3}-\frac{1}{4}a^{2}-\frac{53}{198}a+\frac{1}{3}$, $\frac{1}{6336}a^{14}-\frac{1}{6336}a^{12}+\frac{1}{192}a^{10}+\frac{1}{264}a^{8}-\frac{23}{264}a^{6}-\frac{1}{2}a^{5}-\frac{1}{24}a^{4}-\frac{1}{2}a^{3}-\frac{53}{198}a^{2}-\frac{1}{2}a+\frac{43}{99}$, $\frac{1}{12672}a^{15}-\frac{1}{12672}a^{13}+\frac{1}{384}a^{11}+\frac{1}{528}a^{9}-\frac{23}{528}a^{7}-\frac{1}{4}a^{6}+\frac{23}{48}a^{5}+\frac{1}{4}a^{4}-\frac{53}{396}a^{3}-\frac{1}{4}a^{2}+\frac{43}{198}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{113}{12672} a^{15} + \frac{27}{1408} a^{13} - \frac{31}{384} a^{11} + \frac{73}{264} a^{9} + \frac{1663}{1056} a^{7} - \frac{13}{24} a^{5} - \frac{593}{792} a^{3} + \frac{103}{66} a \) (order $12$) | 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: | $\frac{1}{132}a^{15}+\frac{13}{792}a^{13}-\frac{1}{16}a^{11}+\frac{43}{176}a^{9}+\frac{227}{176}a^{7}-\frac{1}{4}a^{5}+\frac{5}{33}a^{3}+\frac{197}{198}a+1$, $\frac{7}{1584}a^{15}-\frac{17}{3168}a^{14}+\frac{61}{6336}a^{13}-\frac{31}{3168}a^{12}-\frac{7}{192}a^{11}+\frac{5}{96}a^{10}+\frac{301}{2112}a^{9}-\frac{101}{528}a^{8}+\frac{827}{1056}a^{7}-\frac{487}{528}a^{6}-\frac{7}{48}a^{5}+\frac{2}{3}a^{4}+\frac{35}{396}a^{3}+\frac{7}{396}a^{2}+\frac{199}{198}a-\frac{106}{99}$, $\frac{1}{352}a^{15}+\frac{1}{3168}a^{14}+\frac{23}{3168}a^{13}-\frac{7}{3168}a^{12}-\frac{1}{48}a^{11}-\frac{1}{96}a^{10}+\frac{83}{1056}a^{9}+\frac{5}{176}a^{8}+\frac{569}{1056}a^{7}-\frac{7}{132}a^{6}+\frac{5}{48}a^{5}-\frac{1}{2}a^{4}-\frac{73}{264}a^{3}-\frac{47}{396}a^{2}+\frac{103}{198}a+\frac{41}{99}$, $\frac{91}{12672}a^{15}-\frac{19}{3168}a^{14}+\frac{17}{1408}a^{13}-\frac{13}{1056}a^{12}-\frac{25}{384}a^{11}+\frac{5}{96}a^{10}+\frac{281}{1056}a^{9}-\frac{49}{264}a^{8}+\frac{593}{528}a^{7}-\frac{541}{528}a^{6}-\frac{37}{48}a^{5}+\frac{7}{24}a^{4}+\frac{193}{396}a^{3}+\frac{133}{198}a^{2}+\frac{95}{66}a-\frac{16}{11}$, $\frac{1}{352}a^{15}+\frac{1}{288}a^{14}+\frac{19}{2112}a^{13}+\frac{5}{528}a^{12}-\frac{1}{64}a^{11}-\frac{1}{48}a^{10}+\frac{155}{2112}a^{9}+\frac{3}{32}a^{8}+\frac{197}{352}a^{7}+\frac{85}{132}a^{6}+\frac{19}{48}a^{5}+\frac{3}{8}a^{4}+\frac{2}{11}a^{3}+\frac{7}{36}a^{2}+\frac{19}{33}a-\frac{2}{33}$, $\frac{7}{12672}a^{15}-\frac{5}{1152}a^{13}-\frac{5}{384}a^{11}+\frac{23}{352}a^{9}-\frac{11}{96}a^{7}-\frac{3}{4}a^{5}+\frac{545}{792}a^{3}-\frac{13}{18}a$, $\frac{31}{6336}a^{15}+\frac{31}{6336}a^{14}+\frac{7}{576}a^{13}+\frac{59}{6336}a^{12}-\frac{3}{64}a^{11}-\frac{7}{192}a^{10}+\frac{31}{264}a^{9}+\frac{67}{352}a^{8}+\frac{15}{16}a^{7}+\frac{415}{528}a^{6}-\frac{1}{6}a^{5}-\frac{1}{4}a^{4}-\frac{323}{198}a^{3}+\frac{205}{198}a^{2}+\frac{1}{18}a+\frac{202}{99}$ | 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: | \( 6753.06260878 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6753.06260878 \cdot 3}{12\cdot\sqrt{33672290266555416576}}\cr\approx \mathstrut & 0.706713525913 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 16T9):
A solvable group of order 16 |
The 10 conjugacy class representatives for $D_4\times C_2$ |
Character table for $D_4\times C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.4.5802782976.1, 8.0.644753664.3, 8.0.362673936.1, 8.0.10969344.1 |
Minimal sibling: | 8.0.10969344.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(23\) | 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 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
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 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |