Normalized defining polynomial
\( x^{16} - 2 x^{15} + 2 x^{14} - 2 x^{13} - 5 x^{12} - 2 x^{11} - 9 x^{10} + 32 x^{9} + 48 x^{8} - 14 x^{7} - 17 x^{6} + 6 x^{5} - 13 x^{4} - 10 x^{3} + 5 x^{2} + 4 x + 1 \)
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: | \(285727024783360000=2^{16}\cdot 5^{4}\cdot 17^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.33$ | 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: | $2, 5, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{12} - \frac{1}{2} a^{11} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{190} a^{14} + \frac{1}{190} a^{13} + \frac{29}{190} a^{12} - \frac{1}{38} a^{11} + \frac{49}{190} a^{10} + \frac{5}{38} a^{9} + \frac{7}{190} a^{8} + \frac{7}{190} a^{7} - \frac{1}{19} a^{6} - \frac{14}{95} a^{5} - \frac{9}{95} a^{4} - \frac{37}{95} a^{3} + \frac{17}{190} a^{2} - \frac{5}{38} a + \frac{17}{95}$, $\frac{1}{5125630} a^{15} - \frac{11379}{5125630} a^{14} - \frac{28846}{2562815} a^{13} + \frac{955667}{5125630} a^{12} - \frac{796933}{2562815} a^{11} + \frac{137609}{1025126} a^{10} + \frac{364759}{2562815} a^{9} - \frac{1897801}{5125630} a^{8} + \frac{1211247}{5125630} a^{7} - \frac{26792}{134885} a^{6} + \frac{461848}{2562815} a^{5} - \frac{889759}{2562815} a^{4} + \frac{81573}{269770} a^{3} - \frac{1598611}{5125630} a^{2} + \frac{165165}{1025126} a + \frac{730206}{2562815}$
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: | \( \frac{17105}{19342} a^{15} - \frac{21257}{9671} a^{14} + \frac{60625}{19342} a^{13} - \frac{78997}{19342} a^{12} - \frac{22475}{19342} a^{11} - \frac{57735}{19342} a^{10} - \frac{123067}{19342} a^{9} + \frac{582463}{19342} a^{8} + \frac{246691}{9671} a^{7} - \frac{274397}{19342} a^{6} - \frac{23951}{9671} a^{5} + \frac{33931}{9671} a^{4} - \frac{207617}{19342} a^{3} - \frac{34712}{9671} a^{2} + \frac{2081}{509} a + \frac{27519}{19342} \) (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: | \( 210.317511063 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_8:C_2^2$ (as 16T35):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_8:C_2^2$ |
| Character table for $C_8:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), 4.0.272.1 x2, 4.2.1156.1 x2, \(\Q(i, \sqrt{17})\), 8.0.31443200.1 x2, 8.0.31443200.2 x2, 8.0.21381376.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
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 | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.2.1.1 | $x^{2} - 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} - 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} - 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} - 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |