Normalized defining polynomial
\( x^{16} - x^{15} - 34 x^{13} - 27 x^{12} + 85 x^{11} + 289 x^{10} + 762 x^{9} - 717 x^{8} - 910 x^{7} - 2553 x^{6} - 48 x^{5} + 3235 x^{4} - 436 x^{3} + 3761 x^{2} - 4957 x + 1381 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(462639521059669144140625=5^{8}\cdot 13^{6}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.14$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 13, 19, 29, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{39} a^{14} + \frac{14}{39} a^{13} + \frac{14}{39} a^{12} + \frac{2}{13} a^{11} + \frac{10}{39} a^{10} - \frac{5}{39} a^{9} + \frac{3}{13} a^{8} + \frac{5}{39} a^{7} + \frac{4}{13} a^{6} + \frac{2}{13} a^{5} - \frac{6}{13} a^{4} - \frac{4}{13} a^{3} - \frac{8}{39} a^{2} + \frac{2}{39} a + \frac{16}{39}$, $\frac{1}{6750668189216460641252055819} a^{15} - \frac{77604611955783237244332316}{6750668189216460641252055819} a^{14} - \frac{3064150447862197514939246902}{6750668189216460641252055819} a^{13} - \frac{1118218370294293797618567548}{2250222729738820213750685273} a^{12} - \frac{2723202962665040814220483064}{6750668189216460641252055819} a^{11} + \frac{3135682217137682207172518425}{6750668189216460641252055819} a^{10} + \frac{670696270657148952929256725}{2250222729738820213750685273} a^{9} - \frac{2951164594652395158475641127}{6750668189216460641252055819} a^{8} - \frac{874182242315428267489817008}{2250222729738820213750685273} a^{7} + \frac{499306921919120752063455279}{2250222729738820213750685273} a^{6} + \frac{323739222984349763600041179}{2250222729738820213750685273} a^{5} + \frac{958332234945763684073835396}{2250222729738820213750685273} a^{4} + \frac{1755779275261080747766740727}{6750668189216460641252055819} a^{3} - \frac{1296168941977720221337550077}{6750668189216460641252055819} a^{2} - \frac{183147387023423572025684906}{519282168401266203173235063} a + \frac{1044053263348324881070251201}{2250222729738820213750685273}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 639016.829133 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 16384 |
| The 148 conjugacy class representatives for t16n1772 are not computed |
| Character table for t16n1772 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.309593125.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.8.0.1 | $x^{8} - x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $29$ | 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 31 | Data not computed | ||||||