Normalized defining polynomial
\( x^{16} - 6x^{8} + 18 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(11293466247102272049053171712\) \(\medspace = 2^{71}\cdot 3^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.66\) | 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: | $2^{319/64}3^{7/8}\approx 82.78039273967222$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1}{3} a^{8} - 1 \) (order $4$) | 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: | $\frac{1}{3}a^{12}-\frac{2}{3}a^{8}+a^{4}+3$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{12}+a^{10}+\frac{1}{3}a^{8}+3a^{4}-2a^{2}-5$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-2a^{6}-3a^{4}-4a^{2}-5$, $\frac{1}{3}a^{15}-\frac{2}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{4}{3}a^{11}+a^{10}+\frac{1}{3}a^{9}-\frac{5}{3}a^{8}+2a^{6}+4a^{5}+2a^{4}-2a^{3}-6a^{2}-6a-1$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-a^{12}-\frac{4}{3}a^{11}-a^{10}+\frac{1}{3}a^{9}+2a^{8}+2a^{6}+4a^{5}+4a^{4}+2a^{3}-6a-11$, $17a^{15}-\frac{61}{3}a^{14}+\frac{71}{3}a^{13}-\frac{80}{3}a^{12}+\frac{91}{3}a^{11}-34a^{10}+39a^{9}-\frac{127}{3}a^{8}-58a^{7}+75a^{6}-97a^{5}+119a^{4}-146a^{3}+176a^{2}-216a+263$, $\frac{65}{3}a^{15}+\frac{8}{3}a^{14}-\frac{70}{3}a^{13}-\frac{5}{3}a^{12}+37a^{11}+\frac{22}{3}a^{10}-64a^{9}-\frac{131}{3}a^{8}-64a^{7}+59a^{6}+79a^{5}-78a^{4}-116a^{3}+130a^{2}+258a-79$ (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)]
| |
Regulator: | \( 42524370.28488557 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 42524370.28488557 \cdot 2}{4\cdot\sqrt{11293466247102272049053171712}}\cr\approx \mathstrut & 0.485996580157669 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 128 |
The 32 conjugacy class representatives for $D_8:C_8$ |
Character table for $D_8:C_8$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.4608.1, 8.0.97844723712.9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | 16.0.11293466247102272049053171712.7 |
Degree 32 siblings: | deg 32, deg 32 |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | $16$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.71.31 | $x^{16} + 16 x^{14} + 58 x^{8} + 32 x^{6} + 16 x^{4} + 2$ | $16$ | $1$ | $71$ | 16T260 | $[2, 3, 7/2, 4, 9/2, 5, 11/2]$ |
\(3\) | 3.16.14.5 | $x^{16} - 6 x^{8} + 18$ | $8$ | $2$ | $14$ | 16T124 | $[\ ]_{8}^{8}$ |