Normalized defining polynomial
\( x^{16} - x^{15} + 3x^{13} - 6x^{11} + 5x^{10} + 4x^{9} - 3x^{8} + 4x^{7} + 5x^{6} - 6x^{5} + 3x^{3} - x + 1 \)
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: | \(9227446944279201\) \(\medspace = 3^{16}\cdot 11^{8}\) | 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: | \(9.95\) | 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: | $3^{7/6}11^{2/3}\approx 17.819817582388065$ | ||
Ramified primes: | \(3\), \(11\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{4}{9}a^{6}+\frac{2}{9}a^{5}+\frac{1}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{1}{3}a^{2}-\frac{1}{9}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{10}-\frac{1}{3}a^{5}-\frac{2}{9}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{2}{9}a^{4}-\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{9}+\frac{1}{9}a^{6}-\frac{1}{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( \frac{5}{9} a^{15} - \frac{7}{9} a^{14} + \frac{1}{9} a^{13} + \frac{14}{9} a^{12} - \frac{5}{9} a^{11} - \frac{31}{9} a^{10} + \frac{29}{9} a^{9} + \frac{5}{3} a^{8} - \frac{5}{3} a^{7} + \frac{17}{9} a^{6} + \frac{2}{9} a^{5} - \frac{35}{9} a^{4} - \frac{1}{9} a^{3} + \frac{7}{9} a^{2} + \frac{2}{9} a + \frac{2}{9} \) (order $6$) | 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: | $\frac{4}{9}a^{15}-\frac{1}{9}a^{14}-\frac{1}{9}a^{13}+a^{12}+a^{11}-\frac{19}{9}a^{10}+\frac{1}{9}a^{9}+\frac{14}{9}a^{8}+\frac{13}{9}a^{7}+\frac{5}{3}a^{6}+\frac{8}{3}a^{5}-\frac{8}{9}a^{4}-\frac{8}{9}a^{3}-\frac{13}{9}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{9}a^{14}-\frac{2}{3}a^{13}+\frac{4}{3}a^{12}+\frac{10}{9}a^{11}-\frac{23}{9}a^{10}-\frac{7}{9}a^{9}+\frac{41}{9}a^{8}-\frac{8}{9}a^{7}-\frac{1}{9}a^{6}+\frac{35}{9}a^{5}-\frac{10}{9}a^{4}-\frac{23}{9}a^{3}+\frac{5}{3}a^{2}+\frac{5}{9}a-\frac{8}{9}$, $\frac{2}{9}a^{15}-\frac{5}{9}a^{14}-\frac{2}{9}a^{13}+\frac{10}{9}a^{12}-\frac{10}{9}a^{11}-\frac{25}{9}a^{10}+\frac{8}{3}a^{9}+2a^{8}-4a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{47}{9}a^{4}-\frac{8}{9}a^{3}+\frac{14}{9}a^{2}-\frac{7}{9}a-1$, $\frac{1}{3}a^{15}-\frac{1}{9}a^{14}-\frac{1}{3}a^{13}+\frac{11}{9}a^{12}+\frac{1}{3}a^{11}-\frac{20}{9}a^{10}+\frac{2}{3}a^{9}+\frac{23}{9}a^{8}-\frac{14}{9}a^{7}+\frac{14}{9}a^{6}+3a^{5}-\frac{19}{9}a^{4}-\frac{5}{3}a^{3}+\frac{2}{9}a^{2}+\frac{2}{9}a-\frac{1}{9}$, $\frac{2}{9}a^{14}+\frac{1}{9}a^{13}-\frac{4}{9}a^{12}+\frac{8}{9}a^{11}+\frac{7}{9}a^{10}-\frac{13}{9}a^{9}-\frac{5}{9}a^{8}+\frac{26}{9}a^{7}-\frac{2}{9}a^{6}+\frac{7}{9}a^{5}+\frac{23}{9}a^{4}-\frac{8}{9}a^{3}-\frac{4}{3}a^{2}+\frac{5}{3}a+1$, $\frac{2}{9}a^{15}-\frac{5}{9}a^{14}+\frac{4}{9}a^{13}+\frac{1}{3}a^{12}-\frac{8}{9}a^{11}-a^{10}+\frac{7}{3}a^{9}-\frac{13}{9}a^{8}-\frac{5}{9}a^{7}+\frac{13}{9}a^{6}-\frac{10}{9}a^{5}-\frac{5}{3}a^{4}+\frac{4}{9}a^{3}-\frac{5}{3}a^{2}-\frac{8}{9}a-\frac{4}{9}$, $\frac{4}{9}a^{15}-\frac{1}{3}a^{13}+\frac{10}{9}a^{12}+\frac{10}{9}a^{11}-\frac{19}{9}a^{10}-\frac{7}{9}a^{9}+\frac{22}{9}a^{8}+\frac{11}{9}a^{7}+\frac{20}{9}a^{6}+\frac{20}{9}a^{5}-\frac{8}{9}a^{4}-\frac{7}{3}a^{3}-\frac{4}{9}a^{2}+\frac{4}{9}a+\frac{2}{3}$ | 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: | \( 44.5181753858 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 44.5181753858 \cdot 1}{6\cdot\sqrt{9227446944279201}}\cr\approx \mathstrut & 0.187622221662 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 16T66):
A solvable group of order 48 |
The 8 conjugacy class representatives for $\GL(2,3)$ |
Character table for $\GL(2,3)$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.2.3267.1, 8.2.32019867.1, 8.2.32019867.2, 8.0.96059601.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | 8.2.32019867.1, 8.2.32019867.2 |
Degree 24 sibling: | data not computed |
Minimal sibling: | 8.2.32019867.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.12.14.6 | $x^{12} + 6 x^{8} + 15 x^{6} + 9 x^{4} + 18 x^{2} + 9$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.6.4.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
11.6.4.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |