Normalized defining polynomial
\( x^{16} - 4 x^{15} + 10 x^{14} - 12 x^{13} + 4 x^{12} + 10 x^{11} + 16 x^{10} - 108 x^{9} + 222 x^{8} - 232 x^{7} + 146 x^{6} - 50 x^{5} + 19 x^{4} + 2 x^{3} + 4 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(224296960000000000=2^{24}\cdot 5^{10}\cdot 37^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.15$ | 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: | $2, 5, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} + \frac{14}{29} a^{11} - \frac{12}{29} a^{10} - \frac{3}{29} a^{8} + \frac{3}{29} a^{7} + \frac{11}{29} a^{6} + \frac{10}{29} a^{5} + \frac{11}{29} a^{3} + \frac{5}{29} a^{2} - \frac{2}{29} a - \frac{9}{29}$, $\frac{1}{29} a^{13} - \frac{5}{29} a^{11} - \frac{6}{29} a^{10} - \frac{3}{29} a^{9} - \frac{13}{29} a^{8} - \frac{2}{29} a^{7} + \frac{1}{29} a^{6} + \frac{5}{29} a^{5} + \frac{11}{29} a^{4} - \frac{4}{29} a^{3} - \frac{14}{29} a^{2} - \frac{10}{29} a + \frac{10}{29}$, $\frac{1}{841} a^{14} - \frac{11}{841} a^{13} - \frac{5}{841} a^{12} + \frac{165}{841} a^{11} + \frac{63}{841} a^{10} + \frac{136}{841} a^{9} + \frac{373}{841} a^{8} - \frac{6}{841} a^{7} + \frac{226}{841} a^{6} + \frac{217}{841} a^{5} + \frac{397}{841} a^{4} - \frac{260}{841} a^{3} + \frac{318}{841} a^{2} - \frac{170}{841} a - \frac{342}{841}$, $\frac{1}{317057} a^{15} + \frac{181}{317057} a^{14} + \frac{24}{10933} a^{13} - \frac{2448}{317057} a^{12} + \frac{137506}{317057} a^{11} - \frac{79002}{317057} a^{10} - \frac{10774}{24389} a^{9} - \frac{44100}{317057} a^{8} - \frac{56925}{317057} a^{7} - \frac{145007}{317057} a^{6} + \frac{79964}{317057} a^{5} - \frac{31858}{317057} a^{4} + \frac{86640}{317057} a^{3} + \frac{10981}{24389} a^{2} - \frac{3735}{24389} a + \frac{26121}{317057}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{8959}{24389} a^{15} + \frac{36931}{24389} a^{14} - \frac{3367}{841} a^{13} + \frac{130940}{24389} a^{12} - \frac{79687}{24389} a^{11} - \frac{54329}{24389} a^{10} - \frac{138720}{24389} a^{9} + \frac{953654}{24389} a^{8} - \frac{2177432}{24389} a^{7} + \frac{2665405}{24389} a^{6} - \frac{2206008}{24389} a^{5} + \frac{1223739}{24389} a^{4} - \frac{585360}{24389} a^{3} + \frac{103096}{24389} a^{2} - \frac{36413}{24389} a - \frac{33830}{24389} \) (order $4$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 166.240724578 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2.C_2\wr C_2^2$ (as 16T400):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2.C_2\wr C_2^2$ |
| Character table for $C_2.C_2\wr C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.400.1 x2, 4.0.320.1 x2, \(\Q(i, \sqrt{5})\), 8.0.2560000.1 |
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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | Data not computed | ||||||
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $37$ | 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.2 | $x^{4} - 37 x^{2} + 6845$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |