Normalized defining polynomial
\( x^{16} - 72x^{12} + 1764x^{8} + 3024x^{6} - 42768x^{4} + 54432x^{2} + 46818 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(321236373250909071617512439808\) \(\medspace = 2^{79}\cdot 3^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(69.85\) | 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: | $2^{2849/512}3^{3/4}\approx 107.8718783637264$ | ||
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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{9}a^{10}$, $\frac{1}{9}a^{11}$, $\frac{1}{189}a^{12}-\frac{2}{63}a^{8}-\frac{2}{21}a^{4}+\frac{3}{7}$, $\frac{1}{189}a^{13}-\frac{2}{63}a^{9}-\frac{2}{21}a^{5}+\frac{3}{7}a$, $\frac{1}{5753907117}a^{14}+\frac{2595043}{5753907117}a^{12}-\frac{45631847}{1917969039}a^{10}+\frac{81979465}{1917969039}a^{8}-\frac{32490173}{639323013}a^{6}+\frac{96726995}{639323013}a^{4}+\frac{3100560}{6874441}a^{2}+\frac{5341611}{213107671}$, $\frac{1}{97816420989}a^{15}-\frac{10592275}{5753907117}a^{13}-\frac{1324277873}{32605473663}a^{11}+\frac{38847916}{1917969039}a^{9}-\frac{1737351541}{10868491221}a^{7}+\frac{225054034}{3622830407}a^{5}+\frac{51221647}{116865497}a^{3}-\frac{1181972556}{3622830407}a$
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)
| |
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)
| |
Fundamental units: | $\frac{349312}{1917969039}a^{14}+\frac{2435570}{5753907117}a^{12}-\frac{24922102}{1917969039}a^{10}-\frac{23744404}{639323013}a^{8}+\frac{54815800}{213107671}a^{6}+\frac{893602303}{639323013}a^{4}-\frac{29843372}{6874441}a^{2}-\frac{580720415}{213107671}$, $\frac{126748}{5753907117}a^{14}+\frac{13984}{1917969039}a^{12}-\frac{1050872}{639323013}a^{10}-\frac{1036751}{1917969039}a^{8}+\frac{28187000}{639323013}a^{6}+\frac{17171116}{213107671}a^{4}-\frac{8264208}{6874441}a^{2}-\frac{65447645}{213107671}$, $\frac{20960}{213107671}a^{14}-\frac{885793}{5753907117}a^{12}+\frac{13497802}{1917969039}a^{10}+\frac{8527810}{639323013}a^{8}-\frac{30383240}{213107671}a^{6}-\frac{387090139}{639323013}a^{4}+\frac{18401168}{6874441}a^{2}+\frac{190479733}{213107671}$, $\frac{979715}{213107671}a^{14}-\frac{12464477}{821986731}a^{12}+\frac{539308157}{1917969039}a^{10}+\frac{28128490}{30443953}a^{8}-\frac{1079774387}{213107671}a^{6}-\frac{930243371}{30443953}a^{4}+\frac{663138646}{6874441}a^{2}+\frac{2011102325}{30443953}$, $\frac{1822420}{5753907117}a^{14}+\frac{3273181}{5753907117}a^{12}-\frac{7826141}{639323013}a^{10}-\frac{95006195}{1917969039}a^{8}+\frac{159770335}{639323013}a^{6}+\frac{664011253}{639323013}a^{4}-\frac{20466924}{6874441}a^{2}-\frac{434576143}{213107671}$, $\frac{90032360}{32605473663}a^{15}+\frac{43162843}{5753907117}a^{14}-\frac{26300335}{1917969039}a^{13}+\frac{9234734}{273995577}a^{12}+\frac{4442708272}{32605473663}a^{11}-\frac{760977167}{1917969039}a^{10}+\frac{1386509392}{1917969039}a^{9}-\frac{169428277}{91331859}a^{8}-\frac{15175503250}{10868491221}a^{7}+\frac{1067354776}{213107671}a^{6}-\frac{185539523912}{10868491221}a^{5}+\frac{4356235021}{91331859}a^{4}+\frac{3529333968}{116865497}a^{3}-\frac{694303558}{6874441}a^{2}+\frac{83002387438}{3622830407}a-\frac{2209662021}{30443953}$, $\frac{1061327441}{97816420989}a^{15}+\frac{115185199}{5753907117}a^{14}+\frac{210652298}{5753907117}a^{13}+\frac{366888229}{5753907117}a^{12}-\frac{7227258833}{10868491221}a^{11}-\frac{2338044743}{1917969039}a^{10}-\frac{4333086778}{1917969039}a^{9}-\frac{7555180789}{1917969039}a^{8}+\frac{132031183510}{10868491221}a^{7}+\frac{13816495253}{639323013}a^{6}+\frac{266926208074}{3622830407}a^{5}+\frac{85948278925}{639323013}a^{4}-\frac{27049735658}{116865497}a^{3}-\frac{2901924722}{6874441}a^{2}-\frac{577680535874}{3622830407}a-\frac{60965760839}{213107671}$, $\frac{84100288}{5753907117}a^{15}+\frac{51801352}{1917969039}a^{14}-\frac{297801284}{5753907117}a^{13}+\frac{60926870}{639323013}a^{12}+\frac{1715858464}{1917969039}a^{11}-\frac{1082125144}{639323013}a^{10}+\frac{6229570964}{1917969039}a^{9}-\frac{1229301522}{213107671}a^{8}-\frac{10635463708}{639323013}a^{7}+\frac{20861218610}{639323013}a^{6}-\frac{22148619376}{213107671}a^{5}+\frac{37661873498}{213107671}a^{4}+\frac{2220889042}{6874441}a^{3}-\frac{3952431634}{6874441}a^{2}+\frac{47256886994}{213107671}a-\frac{82843812083}{213107671}$, $\frac{1329406775}{13973774427}a^{15}+\frac{142135291}{639323013}a^{14}+\frac{1000425340}{1917969039}a^{13}+\frac{6959356717}{5753907117}a^{12}-\frac{18845318222}{4657924809}a^{11}-\frac{2031600586}{213107671}a^{10}-\frac{43465882315}{1917969039}a^{9}-\frac{33737868095}{639323013}a^{8}+\frac{23050864478}{517547201}a^{7}+\frac{22768996142}{213107671}a^{6}+\frac{5959214941310}{10868491221}a^{5}+\frac{821014277017}{639323013}a^{4}-\frac{17357119450}{16695071}a^{3}-\frac{17034860282}{6874441}a^{2}-\frac{2838894345152}{3622830407}a-\frac{396376858107}{213107671}$ (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: | \( 520638601.5089489 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 520638601.5089489 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.452161108420426 \end{aligned}\] (assuming GRH)
Galois group
$C_2^7.C_8$ (as 16T1155):
A solvable group of order 1024 |
The 40 conjugacy class representatives for $C_2^7.C_8$ |
Character table for $C_2^7.C_8$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.173946175488.2 |
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 | R | $16$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | $16$ | $16$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $16$ | $16$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.79.2533 | $x^{16} + 40 x^{12} + 32 x^{11} + 32 x^{9} + 36 x^{8} + 32 x^{7} + 16 x^{6} + 32 x^{5} + 2$ | $16$ | $1$ | $79$ | 16T1155 | $[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$ |
\(3\) | 3.16.12.3 | $x^{16} - 6 x^{12} + 162$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |