Normalized defining polynomial
\( x^{16} - x^{15} + 2 x^{13} - 4 x^{12} - x^{11} + 7 x^{10} - 12 x^{9} + 8 x^{8} + 12 x^{7} + 7 x^{6} + \cdots + 1 \)
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: | \(52333711181640625\) \(\medspace = 5^{12}\cdot 11^{8}\) | 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: | \(11.09\) | 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: | $5^{3/4}11^{1/2}\approx 11.08980336897316$ | ||
Ramified primes: | \(5\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a$, $\frac{1}{20}a^{12}+\frac{1}{20}a^{11}-\frac{1}{10}a^{10}+\frac{1}{10}a^{9}+\frac{1}{5}a^{7}-\frac{1}{10}a^{6}-\frac{1}{5}a^{5}-\frac{1}{2}a^{4}+\frac{2}{5}a^{3}+\frac{3}{20}a^{2}+\frac{9}{20}a+\frac{3}{10}$, $\frac{1}{20}a^{13}+\frac{1}{10}a^{11}-\frac{1}{20}a^{10}-\frac{1}{10}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{10}a^{6}+\frac{1}{5}a^{5}-\frac{1}{10}a^{4}-\frac{1}{4}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{9}{20}$, $\frac{1}{20}a^{14}+\frac{1}{10}a^{11}+\frac{1}{10}a^{10}+\frac{1}{5}a^{8}-\frac{1}{10}a^{6}-\frac{1}{5}a^{5}-\frac{1}{4}a^{4}-\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{10}$, $\frac{1}{200}a^{15}+\frac{1}{100}a^{14}-\frac{1}{50}a^{13}-\frac{7}{100}a^{11}-\frac{3}{200}a^{10}-\frac{3}{50}a^{9}+\frac{3}{50}a^{8}+\frac{11}{50}a^{7}+\frac{11}{50}a^{6}-\frac{41}{200}a^{5}+\frac{49}{100}a^{4}-\frac{2}{5}a^{3}+\frac{7}{50}a^{2}-\frac{23}{100}a+\frac{83}{200}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{9}{40} a^{15} - \frac{9}{20} a^{14} + \frac{3}{10} a^{13} + \frac{9}{20} a^{12} - \frac{27}{20} a^{11} + \frac{27}{40} a^{10} + \frac{9}{5} a^{9} - \frac{23}{5} a^{8} + \frac{9}{2} a^{7} + \frac{9}{10} a^{6} - \frac{9}{8} a^{5} - \frac{27}{20} a^{4} + \frac{21}{10} a^{3} + \frac{9}{20} a^{2} + \frac{9}{20} a + \frac{9}{40} \) (order $10$) | 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: | $\frac{7}{100}a^{15}-\frac{3}{50}a^{14}-\frac{2}{25}a^{13}+\frac{1}{50}a^{11}-\frac{31}{100}a^{10}+\frac{13}{50}a^{9}+\frac{17}{50}a^{8}-\frac{3}{25}a^{7}+\frac{2}{25}a^{6}+\frac{323}{100}a^{5}-\frac{76}{25}a^{4}-\frac{31}{10}a^{3}-\frac{1}{25}a^{2}-\frac{1}{50}a+\frac{101}{100}$, $\frac{1}{100}a^{15}-\frac{2}{25}a^{14}+\frac{21}{100}a^{13}-\frac{1}{5}a^{12}-\frac{1}{25}a^{11}+\frac{21}{50}a^{10}-\frac{13}{25}a^{9}-\frac{7}{25}a^{8}+\frac{41}{25}a^{7}-\frac{123}{50}a^{6}+\frac{179}{100}a^{5}-\frac{13}{25}a^{4}+\frac{27}{20}a^{3}+\frac{2}{25}a^{2}-\frac{43}{50}a-\frac{21}{50}$, $\frac{1}{100}a^{15}-\frac{2}{25}a^{14}-\frac{7}{50}a^{13}+\frac{3}{10}a^{12}-\frac{6}{25}a^{11}-\frac{23}{100}a^{10}+\frac{59}{50}a^{9}-\frac{17}{25}a^{8}-\frac{19}{25}a^{7}+\frac{137}{50}a^{6}-\frac{311}{100}a^{5}-\frac{141}{50}a^{4}+\frac{1}{10}a^{3}-\frac{1}{50}a^{2}+\frac{47}{50}a-\frac{107}{100}$, $\frac{2}{25}a^{15}-\frac{29}{100}a^{14}+\frac{43}{100}a^{13}-\frac{7}{20}a^{12}-\frac{37}{100}a^{11}+\frac{53}{50}a^{10}-\frac{33}{50}a^{9}-\frac{67}{50}a^{8}+\frac{231}{50}a^{7}-\frac{269}{50}a^{6}+\frac{98}{25}a^{5}-\frac{241}{100}a^{4}-\frac{29}{20}a^{3}-\frac{51}{100}a^{2}-\frac{53}{100}a-\frac{3}{50}$, $\frac{37}{200}a^{15}-\frac{43}{100}a^{14}+\frac{21}{100}a^{13}+\frac{3}{10}a^{12}-\frac{31}{25}a^{11}+\frac{159}{200}a^{10}+\frac{37}{25}a^{9}-\frac{92}{25}a^{8}+\frac{116}{25}a^{7}+\frac{6}{25}a^{6}-\frac{257}{200}a^{5}-\frac{177}{100}a^{4}-\frac{53}{20}a^{3}-\frac{101}{50}a^{2}-\frac{24}{25}a+\frac{61}{200}$, $\frac{13}{200}a^{15}+\frac{2}{25}a^{14}-\frac{21}{100}a^{13}+\frac{7}{20}a^{12}-\frac{3}{50}a^{11}-\frac{159}{200}a^{10}+\frac{41}{50}a^{9}-\frac{11}{50}a^{8}-\frac{51}{25}a^{7}+\frac{183}{50}a^{6}+\frac{67}{200}a^{5}+\frac{51}{50}a^{4}+\frac{27}{20}a^{3}+\frac{137}{100}a^{2}-\frac{1}{25}a-\frac{41}{200}$, $\frac{19}{100}a^{15}-\frac{11}{50}a^{14}-\frac{4}{25}a^{13}+\frac{3}{5}a^{12}-\frac{81}{100}a^{11}-\frac{57}{100}a^{10}+\frac{111}{50}a^{9}-\frac{111}{50}a^{8}+\frac{4}{25}a^{7}+\frac{233}{50}a^{6}-\frac{39}{100}a^{5}-\frac{77}{25}a^{4}-\frac{9}{10}a^{3}+\frac{3}{25}a^{2}-\frac{49}{100}a+\frac{97}{100}$ | 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: | \( 190.616473208 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 190.616473208 \cdot 1}{10\cdot\sqrt{52333711181640625}}\cr\approx \mathstrut & 0.202399115453 \end{aligned}\]
Galois group
$C_2^2:C_4$ (as 16T10):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_2^2 : C_4$ |
Character table for $C_2^2 : C_4$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.0.1890625.1, 8.4.228765625.1 |
Minimal sibling: | 8.0.1890625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
\(11\) | 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |