Normalized defining polynomial
\( x^{16} - 4 x^{15} + 14 x^{14} - 36 x^{13} + 80 x^{12} - 152 x^{11} + 256 x^{10} - 372 x^{9} + 451 x^{8} + \cdots + 1 \)
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: | \(20542695432781824\) \(\medspace = 2^{32}\cdot 3^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.46\) | 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: | $2^{2}3^{7/8}\approx 10.460226514460828$ | ||
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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $16$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11}a^{14}-\frac{3}{11}a^{13}-\frac{2}{11}a^{12}+\frac{1}{11}a^{11}-\frac{3}{11}a^{10}-\frac{3}{11}a^{9}-\frac{5}{11}a^{8}+\frac{3}{11}a^{7}+\frac{2}{11}a^{6}+\frac{3}{11}a^{5}-\frac{3}{11}a^{4}-\frac{5}{11}a^{3}+\frac{4}{11}a^{2}-\frac{1}{11}a-\frac{5}{11}$, $\frac{1}{4830529}a^{15}-\frac{199241}{4830529}a^{14}+\frac{1965782}{4830529}a^{13}-\frac{1608440}{4830529}a^{12}+\frac{2149083}{4830529}a^{11}-\frac{515819}{4830529}a^{10}+\frac{1165023}{4830529}a^{9}+\frac{120984}{439139}a^{8}+\frac{2308224}{4830529}a^{7}+\frac{1618245}{4830529}a^{6}+\frac{30295}{210023}a^{5}-\frac{1219425}{4830529}a^{4}-\frac{1268640}{4830529}a^{3}+\frac{1719429}{4830529}a^{2}+\frac{214809}{4830529}a+\frac{2342755}{4830529}$
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: | \( \frac{116163748}{4830529} a^{15} - \frac{400145791}{4830529} a^{14} + \frac{1400761486}{4830529} a^{13} - \frac{3394148708}{4830529} a^{12} + \frac{7372792822}{4830529} a^{11} - \frac{13482010844}{4830529} a^{10} + \frac{22077378896}{4830529} a^{9} - \frac{30647044013}{4830529} a^{8} + \frac{34879441088}{4830529} a^{7} - \frac{31554781848}{4830529} a^{6} + \frac{864222976}{210023} a^{5} - \frac{8354939592}{4830529} a^{4} + \frac{4347436208}{4830529} a^{3} - \frac{3058729513}{4830529} a^{2} + \frac{1248592326}{4830529} a - \frac{189141914}{4830529} \) (order $12$) | 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: | $\frac{28284373}{4830529}a^{15}-\frac{101968435}{4830529}a^{14}+\frac{353546668}{4830529}a^{13}-\frac{871994087}{4830529}a^{12}+\frac{1894700811}{4830529}a^{11}-\frac{3496543957}{4830529}a^{10}+\frac{5742297102}{4830529}a^{9}-\frac{731427200}{439139}a^{8}+\frac{9242484117}{4830529}a^{7}-\frac{8457811095}{4830529}a^{6}+\frac{236838026}{210023}a^{5}-\frac{2298221153}{4830529}a^{4}+\frac{1129579172}{4830529}a^{3}-\frac{833433215}{4830529}a^{2}+\frac{357403812}{4830529}a-\frac{53258655}{4830529}$, $\frac{56144247}{4830529}a^{15}-\frac{206698706}{4830529}a^{14}+\frac{716389370}{4830529}a^{13}-\frac{1781804270}{4830529}a^{12}+\frac{3883838833}{4830529}a^{11}-\frac{7205787106}{4830529}a^{10}+\frac{1080202918}{439139}a^{9}-\frac{16757588916}{4830529}a^{8}+\frac{19433722645}{4830529}a^{7}-\frac{18011277106}{4830529}a^{6}+\frac{517578362}{210023}a^{5}-\frac{475072066}{439139}a^{4}+\frac{2504253172}{4830529}a^{3}-\frac{1796297633}{4830529}a^{2}+\frac{804082222}{4830529}a-\frac{139153713}{4830529}$, $\frac{33105568}{4830529}a^{15}-\frac{128794085}{4830529}a^{14}+\frac{446140646}{4830529}a^{13}-\frac{1132508686}{4830529}a^{12}+\frac{2487867259}{4830529}a^{11}-\frac{4673501927}{4830529}a^{10}+\frac{7778219881}{4830529}a^{9}-\frac{11131515276}{4830529}a^{8}+\frac{13175487803}{4830529}a^{7}-\frac{12530020828}{4830529}a^{6}+\frac{34341569}{19093}a^{5}-\frac{4056903639}{4830529}a^{4}+\frac{1816056054}{4830529}a^{3}-\frac{1251869935}{4830529}a^{2}+\frac{635021128}{4830529}a-\frac{126504394}{4830529}$, $\frac{5528458}{4830529}a^{15}-\frac{14125153}{4830529}a^{14}+\frac{52893659}{4830529}a^{13}-\frac{111727559}{4830529}a^{12}+\frac{240637883}{4830529}a^{11}-\frac{405968504}{4830529}a^{10}+\frac{642173270}{4830529}a^{9}-\frac{73619275}{439139}a^{8}+\frac{816236526}{4830529}a^{7}-\frac{624178597}{4830529}a^{6}+\frac{10404703}{210023}a^{5}-\frac{68115837}{4830529}a^{4}+\frac{97438133}{4830529}a^{3}-\frac{28279045}{4830529}a^{2}-\frac{3698012}{4830529}a+\frac{554243}{4830529}$, $\frac{25815681}{4830529}a^{15}-\frac{113825590}{4830529}a^{14}+\frac{385464945}{4830529}a^{13}-\frac{1020605879}{4830529}a^{12}+\frac{2243382538}{4830529}a^{11}-\frac{4300514497}{4830529}a^{10}+\frac{7200192818}{4830529}a^{9}-\frac{10500772563}{4830529}a^{8}+\frac{12644383282}{4830529}a^{7}-\frac{1116330762}{439139}a^{6}+\frac{383199287}{210023}a^{5}-\frac{4159344657}{4830529}a^{4}+\frac{1751876363}{4830529}a^{3}-\frac{1271023299}{4830529}a^{2}+\frac{663014904}{4830529}a-\frac{135789949}{4830529}$, $\frac{46348155}{4830529}a^{15}-\frac{165839559}{4830529}a^{14}+\frac{578761883}{4830529}a^{13}-\frac{1423577035}{4830529}a^{12}+\frac{3104831768}{4830529}a^{11}-\frac{520739612}{439139}a^{10}+\frac{9433910890}{4830529}a^{9}-\frac{13232890043}{4830529}a^{8}+\frac{15273166451}{4830529}a^{7}-\frac{14065946426}{4830529}a^{6}+\frac{399403842}{210023}a^{5}-\frac{4019593273}{4830529}a^{4}+\frac{1957617560}{4830529}a^{3}-\frac{1381553923}{4830529}a^{2}+\frac{612547479}{4830529}a-\frac{106013609}{4830529}$, $\frac{65832496}{4830529}a^{15}-\frac{225485871}{4830529}a^{14}+\frac{791061421}{4830529}a^{13}-\frac{1913028444}{4830529}a^{12}+\frac{4158228970}{4830529}a^{11}-\frac{7599229558}{4830529}a^{10}+\frac{12447374341}{4830529}a^{9}-\frac{17274391152}{4830529}a^{8}+\frac{19665004385}{4830529}a^{7}-\frac{17811496866}{4830529}a^{6}+\frac{488827257}{210023}a^{5}-\frac{4781579038}{4830529}a^{4}+\frac{2492931185}{4830529}a^{3}-\frac{1713240058}{4830529}a^{2}+\frac{705159627}{4830529}a-\frac{114214064}{4830529}$ | 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: | \( 160.565615792 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 160.565615792 \cdot 1}{12\cdot\sqrt{20542695432781824}}\cr\approx \mathstrut & 0.226767878128 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 4.2.1728.1 x2, 4.0.432.1 x2, 8.0.2985984.1, 8.2.143327232.1 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.2.143327232.1 |
Minimal sibling: | 8.2.143327232.1 |
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 | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.32.1 | $x^{16} + 4 x^{14} + 4 x^{12} - 4 x^{11} - 4 x^{10} + 4 x^{8} + 16 x^{7} + 40 x^{6} + 32 x^{5} - 40 x^{3} - 8 x^{2} + 48 x + 36$ | $8$ | $2$ | $32$ | $QD_{16}$ | $[2, 2, 5/2]^{2}$ |
\(3\) | 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |