Normalized defining polynomial
\( x^{16} + 68x^{8} + 4 \)
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)
| |
Discriminant: | \(18446744073709551616\) \(\medspace = 2^{64}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.00\) | 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))
| |
Galois root discriminant: | $2^{4}\approx 16.0$ | ||
Ramified primes: | \(2\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{48}a^{8}-\frac{1}{2}a^{4}-\frac{7}{24}$, $\frac{1}{48}a^{9}-\frac{1}{2}a^{5}-\frac{7}{24}a$, $\frac{1}{48}a^{10}-\frac{1}{2}a^{6}-\frac{7}{24}a^{2}$, $\frac{1}{48}a^{11}-\frac{1}{2}a^{7}-\frac{7}{24}a^{3}$, $\frac{1}{48}a^{12}-\frac{7}{24}a^{4}$, $\frac{1}{96}a^{13}-\frac{1}{2}a^{7}-\frac{7}{48}a^{5}-\frac{1}{2}a$, $\frac{1}{96}a^{14}-\frac{7}{48}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{96}a^{15}-\frac{7}{48}a^{7}-\frac{1}{2}a^{3}$
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)
| |
Torsion generator: | \( \frac{1}{24} a^{12} - \frac{1}{48} a^{8} + \frac{35}{12} a^{4} - \frac{17}{24} \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{7}{96}a^{14}+\frac{239}{48}a^{6}+\frac{1}{2}a^{2}$, $\frac{7}{96}a^{15}+\frac{7}{96}a^{13}+\frac{1}{48}a^{11}+\frac{239}{48}a^{7}+\frac{239}{48}a^{5}+\frac{29}{24}a^{3}-\frac{1}{2}a$, $\frac{7}{96}a^{15}+\frac{7}{96}a^{13}+\frac{1}{48}a^{11}+\frac{239}{48}a^{7}+\frac{239}{48}a^{5}+\frac{29}{24}a^{3}+\frac{1}{2}a$, $\frac{17}{96}a^{14}-\frac{7}{96}a^{13}+\frac{1}{16}a^{11}-\frac{1}{48}a^{10}+\frac{577}{48}a^{6}-\frac{239}{48}a^{5}+\frac{33}{8}a^{3}-\frac{29}{24}a^{2}-\frac{1}{2}a+1$, $\frac{1}{32}a^{15}+\frac{1}{24}a^{14}-\frac{1}{24}a^{13}-\frac{1}{48}a^{11}+\frac{1}{48}a^{10}-\frac{1}{48}a^{9}+\frac{1}{24}a^{8}+\frac{33}{16}a^{7}+\frac{35}{12}a^{6}-\frac{35}{12}a^{5}-\frac{29}{24}a^{3}+\frac{17}{24}a^{2}+\frac{7}{24}a+\frac{5}{12}$, $\frac{1}{32}a^{15}+\frac{1}{24}a^{14}+\frac{1}{24}a^{13}-\frac{1}{48}a^{12}-\frac{1}{48}a^{11}-\frac{1}{48}a^{10}+\frac{1}{48}a^{9}+\frac{33}{16}a^{7}+\frac{35}{12}a^{6}+\frac{35}{12}a^{5}-\frac{41}{24}a^{4}-\frac{29}{24}a^{3}-\frac{17}{24}a^{2}-\frac{7}{24}a$, $\frac{17}{96}a^{15}-\frac{7}{96}a^{14}+\frac{1}{32}a^{13}-\frac{1}{24}a^{11}+\frac{1}{24}a^{10}-\frac{1}{48}a^{9}+\frac{577}{48}a^{7}-\frac{239}{48}a^{6}+\frac{33}{16}a^{5}-\frac{35}{12}a^{3}+\frac{35}{12}a^{2}-\frac{29}{24}a+1$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 5486.67245904 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5486.67245904 \cdot 1}{8\cdot\sqrt{18446744073709551616}}\cr\approx \mathstrut & 0.387880666573 \end{aligned}\]
Galois group
$\SD_{16}$ (as 16T12):
A solvable group of order 16 |
The 7 conjugacy class representatives for $QD_{16}$ |
Character table for $QD_{16}$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 4.2.2048.1 x2, 4.0.2048.1 x2, 8.0.16777216.2, 8.2.2147483648.2 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.2.2147483648.2 |
Minimal sibling: | 8.2.2147483648.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.64.9 | $x^{16} + 16 x^{15} + 16 x^{13} + 12 x^{12} + 16 x^{10} + 22 x^{8} + 16 x^{5} + 4 x^{4} + 24 x^{2} + 16 x + 2$ | $16$ | $1$ | $64$ | $QD_{16}$ | $[2, 3, 4, 5]$ |