Normalized defining polynomial
\( x^{16} - 4x^{14} - 16x^{12} - 36x^{10} + 14x^{8} + 36x^{6} - 16x^{4} + 4x^{2} + 1 \)
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: | \(6871947673600000000\) \(\medspace = 2^{44}\cdot 5^{8}\) | 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: | \(15.04\) | 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: | $2^{23/8}5^{1/2}\approx 16.403867009982392$ | ||
Ramified primes: | \(2\), \(5\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{3}{8}a^{2}-\frac{3}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{40}a^{12}-\frac{1}{8}a^{8}+\frac{1}{10}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{2}+\frac{9}{40}$, $\frac{1}{80}a^{13}-\frac{1}{80}a^{12}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{16}a^{5}+\frac{1}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{31}{80}a+\frac{31}{80}$, $\frac{1}{400}a^{14}+\frac{3}{400}a^{12}+\frac{3}{80}a^{10}-\frac{31}{400}a^{8}-\frac{53}{400}a^{6}-\frac{19}{80}a^{4}+\frac{69}{400}a^{2}+\frac{187}{400}$, $\frac{1}{400}a^{15}-\frac{1}{200}a^{13}-\frac{1}{80}a^{12}+\frac{3}{80}a^{11}-\frac{3}{200}a^{9}+\frac{1}{16}a^{8}+\frac{27}{400}a^{7}+\frac{1}{5}a^{6}-\frac{7}{40}a^{5}+\frac{1}{16}a^{4}-\frac{131}{400}a^{3}-\frac{1}{2}a^{2}-\frac{29}{200}a+\frac{31}{80}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{17}{200}a^{14}-\frac{49}{200}a^{12}-\frac{69}{40}a^{10}-\frac{927}{200}a^{8}-\frac{501}{200}a^{6}+\frac{157}{40}a^{4}+\frac{373}{200}a^{2}-\frac{121}{200}$, $\frac{23}{200}a^{14}-\frac{101}{200}a^{12}-\frac{33}{20}a^{10}-\frac{86}{25}a^{8}+\frac{601}{200}a^{6}+\frac{113}{40}a^{4}-\frac{469}{100}a^{2}+\frac{49}{50}$, $\frac{23}{200}a^{15}-\frac{101}{200}a^{13}-\frac{33}{20}a^{11}-\frac{86}{25}a^{9}+\frac{601}{200}a^{7}+\frac{113}{40}a^{5}-\frac{469}{100}a^{3}+\frac{49}{50}a$, $\frac{119}{400}a^{14}-\frac{473}{400}a^{12}-\frac{383}{80}a^{10}-\frac{4339}{400}a^{8}+\frac{1573}{400}a^{6}+\frac{849}{80}a^{4}-\frac{1889}{400}a^{2}-\frac{217}{400}$, $\frac{47}{400}a^{15}-\frac{7}{80}a^{14}-\frac{61}{100}a^{13}+\frac{7}{20}a^{12}-\frac{109}{80}a^{11}+\frac{23}{16}a^{10}-\frac{183}{100}a^{9}+\frac{121}{40}a^{8}+\frac{2969}{400}a^{7}-\frac{153}{80}a^{6}+\frac{39}{10}a^{5}-5a^{4}-\frac{3007}{400}a^{3}+\frac{77}{80}a^{2}+\frac{3}{50}a+\frac{21}{40}$, $\frac{47}{400}a^{15}+\frac{7}{80}a^{14}-\frac{61}{100}a^{13}-\frac{7}{20}a^{12}-\frac{109}{80}a^{11}-\frac{23}{16}a^{10}-\frac{183}{100}a^{9}-\frac{121}{40}a^{8}+\frac{2969}{400}a^{7}+\frac{153}{80}a^{6}+\frac{39}{10}a^{5}+5a^{4}-\frac{3007}{400}a^{3}-\frac{77}{80}a^{2}+\frac{3}{50}a-\frac{21}{40}$, $\frac{1}{16}a^{15}+\frac{59}{400}a^{14}-\frac{3}{10}a^{13}-\frac{109}{200}a^{12}-\frac{13}{16}a^{11}-\frac{203}{80}a^{10}-\frac{11}{8}a^{9}-\frac{601}{100}a^{8}+\frac{219}{80}a^{7}+\frac{93}{400}a^{6}+\frac{7}{4}a^{5}+\frac{227}{40}a^{4}-\frac{51}{16}a^{3}-\frac{529}{400}a^{2}+\frac{57}{40}a-\frac{9}{50}$, $\frac{21}{100}a^{15}-\frac{21}{100}a^{14}-\frac{41}{50}a^{13}+\frac{41}{50}a^{12}-\frac{139}{40}a^{11}+\frac{139}{40}a^{10}-\frac{194}{25}a^{9}+\frac{194}{25}a^{8}+\frac{141}{50}a^{7}-\frac{141}{50}a^{6}+\frac{191}{20}a^{5}-\frac{191}{20}a^{4}-\frac{327}{200}a^{3}+\frac{327}{200}a^{2}+\frac{97}{100}a+\frac{3}{100}$, $\frac{91}{400}a^{15}-\frac{3}{50}a^{14}-\frac{337}{400}a^{13}+\frac{17}{100}a^{12}-\frac{307}{80}a^{11}+\frac{49}{40}a^{10}-\frac{3821}{400}a^{9}+\frac{84}{25}a^{8}-\frac{163}{400}a^{7}+\frac{183}{100}a^{6}+\frac{501}{80}a^{5}-\frac{14}{5}a^{4}-\frac{321}{400}a^{3}-\frac{503}{200}a^{2}+\frac{677}{400}a-\frac{7}{100}$ (assuming GRH) | 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: | \( 1265.89495006 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 1265.89495006 \cdot 1}{2\cdot\sqrt{6871947673600000000}}\cr\approx \mathstrut & 0.237698774451 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\wr C_2$ (as 16T46):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
Intermediate fields
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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $8$ | $2$ | $44$ | |||
\(5\) | 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |