Normalized defining polynomial
\( x^{16} - 32x^{12} - 120x^{10} - 222x^{8} - 192x^{6} - 80x^{4} - 24x^{2} + 1 \)
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: | \(149587343098087735296\) \(\medspace = 2^{48}\cdot 3^{12}\) | 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: | \(18.24\) | 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: | $2^{3}3^{3/4}\approx 18.23605645563822$ | ||
Ramified primes: | \(2\), \(3\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}+\frac{1}{8}$, $\frac{1}{8}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{2}a^{5}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{11}-\frac{1}{2}a^{6}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{40}a^{12}-\frac{1}{20}a^{10}-\frac{1}{40}a^{8}-\frac{3}{10}a^{6}-\frac{3}{8}a^{4}-\frac{3}{20}a^{2}+\frac{7}{40}$, $\frac{1}{40}a^{13}-\frac{1}{20}a^{11}-\frac{1}{40}a^{9}-\frac{3}{10}a^{7}-\frac{3}{8}a^{5}-\frac{3}{20}a^{3}+\frac{7}{40}a$, $\frac{1}{7720}a^{14}-\frac{1}{1544}a^{12}-\frac{5}{193}a^{10}+\frac{301}{7720}a^{8}-\frac{2499}{7720}a^{6}-\frac{1}{2}a^{5}+\frac{3039}{7720}a^{4}+\frac{153}{386}a^{2}-\frac{1}{2}a+\frac{309}{7720}$, $\frac{1}{7720}a^{15}-\frac{1}{1544}a^{13}-\frac{5}{193}a^{11}+\frac{301}{7720}a^{9}-\frac{2499}{7720}a^{7}-\frac{1}{2}a^{6}+\frac{3039}{7720}a^{5}+\frac{153}{386}a^{3}-\frac{1}{2}a^{2}+\frac{309}{7720}a$
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: | $\frac{1649}{7720}a^{15}-\frac{359}{3860}a^{13}-\frac{52459}{7720}a^{11}-\frac{87549}{3860}a^{9}-\frac{57883}{1544}a^{7}-\frac{94537}{3860}a^{5}-\frac{49067}{7720}a^{3}-\frac{2063}{772}a$, $\frac{217}{3860}a^{14}-\frac{6}{193}a^{12}-\frac{673}{386}a^{10}-\frac{11249}{1930}a^{8}-\frac{40483}{3860}a^{6}-\frac{7628}{965}a^{4}-\frac{1341}{386}a^{2}-\frac{731}{1930}$, $\frac{1221}{7720}a^{15}+\frac{71}{7720}a^{13}-\frac{39427}{7720}a^{11}-\frac{29635}{1544}a^{9}-\frac{268991}{7720}a^{7}-\frac{226581}{7720}a^{5}-\frac{82631}{7720}a^{3}-\frac{38819}{7720}a$, $\frac{1317}{7720}a^{15}+\frac{1}{7720}a^{14}-\frac{23}{7720}a^{13}-\frac{1}{1544}a^{12}-\frac{42029}{7720}a^{11}-\frac{5}{193}a^{10}-\frac{7865}{386}a^{9}+\frac{301}{7720}a^{8}-\frac{293507}{7720}a^{7}+\frac{5221}{7720}a^{6}-\frac{261007}{7720}a^{5}+\frac{14619}{7720}a^{4}-\frac{121217}{7720}a^{3}+\frac{925}{386}a^{2}-\frac{10057}{1930}a+\frac{4169}{7720}$, $\frac{553}{3860}a^{15}+\frac{719}{7720}a^{14}+\frac{67}{7720}a^{13}-\frac{157}{3860}a^{12}-\frac{17767}{3860}a^{11}-\frac{11491}{3860}a^{10}-\frac{67463}{3860}a^{9}-\frac{38181}{3860}a^{8}-\frac{124359}{3860}a^{7}-\frac{122313}{7720}a^{6}-\frac{208401}{7720}a^{5}-\frac{34117}{3860}a^{4}-\frac{36521}{3860}a^{3}-\frac{223}{3860}a^{2}-\frac{7841}{3860}a-\frac{951}{3860}$, $\frac{553}{3860}a^{15}-\frac{719}{7720}a^{14}+\frac{67}{7720}a^{13}+\frac{157}{3860}a^{12}-\frac{17767}{3860}a^{11}+\frac{11491}{3860}a^{10}-\frac{67463}{3860}a^{9}+\frac{38181}{3860}a^{8}-\frac{124359}{3860}a^{7}+\frac{122313}{7720}a^{6}-\frac{208401}{7720}a^{5}+\frac{34117}{3860}a^{4}-\frac{36521}{3860}a^{3}+\frac{223}{3860}a^{2}-\frac{7841}{3860}a+\frac{951}{3860}$, $\frac{553}{3860}a^{15}-\frac{181}{3860}a^{14}+\frac{67}{7720}a^{13}-\frac{3}{193}a^{12}-\frac{17767}{3860}a^{11}+\frac{2321}{1544}a^{10}-\frac{67463}{3860}a^{9}+\frac{5921}{965}a^{8}-\frac{124359}{3860}a^{7}+\frac{47019}{3860}a^{6}-\frac{208401}{7720}a^{5}+\frac{22673}{1930}a^{4}-\frac{36521}{3860}a^{3}+\frac{7161}{1544}a^{2}-\frac{7841}{3860}a+\frac{734}{965}$, $\frac{509}{1930}a^{15}-\frac{139}{7720}a^{14}-\frac{337}{7720}a^{13}+\frac{251}{3860}a^{12}-\frac{65011}{7720}a^{11}+\frac{4061}{7720}a^{10}-\frac{116831}{3860}a^{9}+\frac{407}{3860}a^{8}-\frac{20791}{386}a^{7}-\frac{17023}{7720}a^{6}-\frac{329241}{7720}a^{5}-\frac{22553}{3860}a^{4}-\frac{121663}{7720}a^{3}-\frac{19847}{7720}a^{2}-\frac{4213}{772}a-\frac{921}{3860}$, $\frac{1317}{7720}a^{15}+\frac{181}{3860}a^{14}-\frac{23}{7720}a^{13}+\frac{3}{193}a^{12}-\frac{42029}{7720}a^{11}-\frac{2321}{1544}a^{10}-\frac{7865}{386}a^{9}-\frac{5921}{965}a^{8}-\frac{293507}{7720}a^{7}-\frac{47019}{3860}a^{6}-\frac{261007}{7720}a^{5}-\frac{22673}{1930}a^{4}-\frac{121217}{7720}a^{3}-\frac{7161}{1544}a^{2}-\frac{10057}{1930}a-\frac{734}{965}$ | 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: | \( 10977.0741836 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 10977.0741836 \cdot 1}{2\cdot\sqrt{149587343098087735296}}\cr\approx \mathstrut & 0.441782361241 \end{aligned}\]
Galois group
$C_4^2:C_2$ (as 16T30):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4^2:C_2$ |
Character table for $C_4^2:C_2$ |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.2.4608.2, 4.2.4608.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.339738624.1, 8.4.764411904.3, 8.4.3057647616.1 |
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 sibling: | 16.0.9349208943630483456.3 |
Minimal sibling: | 16.0.9349208943630483456.3 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.48.13 | $x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{6} + 8 x^{2} + 8 x + 14$ | $16$ | $1$ | $48$ | 16T30 | $[2, 3, 3, 7/2]^{2}$ |
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |