Normalized defining polynomial
\( x^{16} - 2 x^{15} + 3 x^{14} + 8 x^{13} - 16 x^{12} + 21 x^{11} + 16 x^{10} - 38 x^{9} + 57 x^{8} + \cdots + 1 \)
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: | \(66202447602479769\) \(\medspace = 3^{14}\cdot 7^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.25\) | 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: | $3^{7/8}7^{3/4}\approx 11.253940841985125$ | ||
Ramified primes: | \(3\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{3}a^{2}-\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{2115}a^{14}-\frac{46}{2115}a^{13}-\frac{10}{423}a^{12}-\frac{49}{423}a^{11}-\frac{301}{2115}a^{10}-\frac{55}{423}a^{9}-\frac{3}{235}a^{8}-\frac{56}{141}a^{7}+\frac{247}{705}a^{6}+\frac{65}{423}a^{5}-\frac{268}{2115}a^{4}+\frac{1}{9}a^{3}+\frac{122}{423}a^{2}-\frac{286}{2115}a+\frac{538}{2115}$, $\frac{1}{167085}a^{15}-\frac{2}{11139}a^{14}+\frac{913}{55695}a^{13}+\frac{637}{33417}a^{12}-\frac{1879}{18565}a^{11}+\frac{1001}{18565}a^{10}+\frac{12493}{167085}a^{9}+\frac{6626}{55695}a^{8}-\frac{5641}{18565}a^{7}-\frac{59729}{167085}a^{6}-\frac{4622}{18565}a^{5}+\frac{18154}{55695}a^{4}+\frac{5527}{33417}a^{3}-\frac{16582}{55695}a^{2}-\frac{7577}{18565}a-\frac{25232}{167085}$
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: | \( -\frac{84}{395} a^{15} + \frac{30}{79} a^{14} - \frac{186}{395} a^{13} - \frac{470}{237} a^{12} + \frac{1289}{395} a^{11} - \frac{3758}{1185} a^{10} - \frac{6406}{1185} a^{9} + \frac{3468}{395} a^{8} - \frac{3774}{395} a^{7} - \frac{844}{395} a^{6} + \frac{3617}{395} a^{5} - \frac{5448}{395} a^{4} + \frac{124}{237} a^{3} + \frac{1144}{395} a^{2} - \frac{6554}{1185} a + \frac{1334}{1185} \) (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)
| |
Fundamental units: | $\frac{13556}{55695}a^{15}-\frac{4706}{11139}a^{14}+\frac{109462}{167085}a^{13}+\frac{67697}{33417}a^{12}-\frac{59697}{18565}a^{11}+\frac{751792}{167085}a^{10}+\frac{728509}{167085}a^{9}-\frac{399017}{55695}a^{8}+\frac{227432}{18565}a^{7}-\frac{344}{395}a^{6}-\frac{86931}{18565}a^{5}+\frac{2168971}{167085}a^{4}-\frac{31549}{33417}a^{3}+\frac{14043}{18565}a^{2}+\frac{882781}{167085}a-\frac{136301}{167085}$, $\frac{28082}{167085}a^{15}-\frac{3629}{11139}a^{14}+\frac{84358}{167085}a^{13}+\frac{42292}{33417}a^{12}-\frac{133519}{55695}a^{11}+\frac{553348}{167085}a^{10}+\frac{120242}{55695}a^{9}-\frac{88656}{18565}a^{8}+\frac{457334}{55695}a^{7}-\frac{368878}{167085}a^{6}-\frac{88022}{55695}a^{5}+\frac{1210084}{167085}a^{4}-\frac{9038}{33417}a^{3}+\frac{44536}{55695}a^{2}+\frac{6062}{3555}a+\frac{22712}{55695}$, $\frac{28082}{167085}a^{15}-\frac{3629}{11139}a^{14}+\frac{84358}{167085}a^{13}+\frac{42292}{33417}a^{12}-\frac{133519}{55695}a^{11}+\frac{553348}{167085}a^{10}+\frac{120242}{55695}a^{9}-\frac{88656}{18565}a^{8}+\frac{457334}{55695}a^{7}-\frac{368878}{167085}a^{6}-\frac{88022}{55695}a^{5}+\frac{1210084}{167085}a^{4}-\frac{9038}{33417}a^{3}+\frac{44536}{55695}a^{2}+\frac{6062}{3555}a-\frac{32983}{55695}$, $\frac{14578}{167085}a^{15}-\frac{8636}{55695}a^{14}+\frac{4964}{33417}a^{13}+\frac{9787}{11139}a^{12}-\frac{26462}{18565}a^{11}+\frac{31799}{33417}a^{10}+\frac{437174}{167085}a^{9}-\frac{46220}{11139}a^{8}+\frac{187306}{55695}a^{7}+\frac{45146}{33417}a^{6}-\frac{6439}{1185}a^{5}+\frac{209642}{33417}a^{4}-\frac{1594}{3713}a^{3}-\frac{123536}{55695}a^{2}+\frac{460609}{167085}a-\frac{41609}{33417}$, $\frac{12757}{167085}a^{15}-\frac{29738}{167085}a^{14}+\frac{46886}{167085}a^{13}+\frac{2040}{3713}a^{12}-\frac{233477}{167085}a^{11}+\frac{328991}{167085}a^{10}+\frac{150281}{167085}a^{9}-\frac{55927}{18565}a^{8}+\frac{273484}{55695}a^{7}-\frac{269501}{167085}a^{6}-\frac{130916}{167085}a^{5}+\frac{785648}{167085}a^{4}-\frac{7228}{3713}a^{3}+\frac{30853}{167085}a^{2}+\frac{154232}{167085}a-\frac{99928}{167085}$, $\frac{1729}{18565}a^{15}-\frac{15424}{167085}a^{14}+\frac{24463}{167085}a^{13}+\frac{29299}{33417}a^{12}-\frac{88786}{167085}a^{11}+\frac{138718}{167085}a^{10}+\frac{369698}{167085}a^{9}-\frac{12291}{18565}a^{8}+\frac{132517}{55695}a^{7}+\frac{68549}{55695}a^{6}-\frac{12868}{167085}a^{5}+\frac{410659}{167085}a^{4}+\frac{64549}{33417}a^{3}+\frac{398744}{167085}a^{2}+\frac{227236}{167085}a+\frac{40826}{167085}$, $\frac{7486}{55695}a^{15}-\frac{6604}{18565}a^{14}+\frac{97388}{167085}a^{13}+\frac{27523}{33417}a^{12}-\frac{51687}{18565}a^{11}+\frac{691553}{167085}a^{10}+\frac{114239}{167085}a^{9}-\frac{340868}{55695}a^{8}+\frac{562501}{55695}a^{7}-\frac{251836}{55695}a^{6}-\frac{43066}{18565}a^{5}+\frac{1475849}{167085}a^{4}-\frac{133892}{33417}a^{3}-\frac{55756}{55695}a^{2}+\frac{587882}{167085}a-\frac{36469}{167085}$ | 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: | \( 134.265380275 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 134.265380275 \cdot 1}{6\cdot\sqrt{66202447602479769}}\cr\approx \mathstrut & 0.211258756714 \end{aligned}\]
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
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.2.1323.1 x2, 4.0.189.1 x2, 8.0.1750329.1, 8.2.257298363.1 x4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | 8.2.257298363.1 |
Minimal sibling: | 8.2.257298363.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
\(7\) | 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |