Normalized defining polynomial
\( x^{16} - 4 x^{14} - 7 x^{13} + 4 x^{12} + 46 x^{11} + 2 x^{10} - 134 x^{9} + 38 x^{8} + 354 x^{7} - 328 x^{6} + 24 x^{5} + 309 x^{4} - 348 x^{3} + 116 x^{2} - 115 x - 29 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(43527014496875390625=3^{8}\cdot 5^{8}\cdot 19^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.88$ | 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: | $3, 5, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{175} a^{14} + \frac{36}{175} a^{13} - \frac{67}{175} a^{12} - \frac{68}{175} a^{11} + \frac{59}{175} a^{10} + \frac{82}{175} a^{9} - \frac{52}{175} a^{8} - \frac{44}{175} a^{7} - \frac{3}{175} a^{6} + \frac{17}{175} a^{5} - \frac{2}{25} a^{4} + \frac{6}{25} a^{3} + \frac{22}{175} a^{2} + \frac{66}{175} a + \frac{69}{175}$, $\frac{1}{527336416794334813925} a^{15} + \frac{277268540821994247}{105467283358866962785} a^{14} - \frac{15309568322731403029}{75333773827762116275} a^{13} - \frac{16761193019061942993}{75333773827762116275} a^{12} + \frac{122070148364236824252}{527336416794334813925} a^{11} - \frac{85458330499633719177}{527336416794334813925} a^{10} + \frac{114125585906686135141}{527336416794334813925} a^{9} - \frac{267965784327589658}{10761967689680302325} a^{8} + \frac{2421975689290769823}{31019789223196165525} a^{7} - \frac{548240224050652636}{1485454695195309335} a^{6} + \frac{244558781372798104819}{527336416794334813925} a^{5} - \frac{17655367380889920792}{75333773827762116275} a^{4} + \frac{17798916984223320221}{105467283358866962785} a^{3} - \frac{30356706000429567083}{75333773827762116275} a^{2} - \frac{229238935993734927172}{527336416794334813925} a - \frac{41911352602022724944}{527336416794334813925}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 2896.84772251 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times D_8$ (as 16T29):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2\times D_8$ |
| Character table for $C_2\times D_8$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{285}) \), 4.2.475.1, 4.2.4275.1, \(\Q(\sqrt{5}, \sqrt{57})\), 8.2.347236875.1, 8.2.4286875.1, 8.4.6597500625.1 |
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 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |