Normalized defining polynomial
\( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + 366 x^{6} - 344 x^{5} + 286 x^{4} - 184 x^{3} + 84 x^{2} - 24 x + 4 \)
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: | \(2353561680715186176=2^{32}\cdot 3^{8}\cdot 17^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.07$ | 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: | $2, 3, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{14} a^{14} - \frac{1}{14} a^{13} + \frac{3}{7} a^{11} + \frac{3}{14} a^{10} - \frac{1}{2} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{14} a^{6} - \frac{1}{2} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{20109322702} a^{15} - \frac{115766717}{10054661351} a^{14} + \frac{890193632}{10054661351} a^{13} + \frac{206087913}{10054661351} a^{12} + \frac{8240855029}{20109322702} a^{11} - \frac{3873869339}{10054661351} a^{10} - \frac{4615615757}{10054661351} a^{9} + \frac{3559663147}{10054661351} a^{8} + \frac{671326755}{2872760386} a^{7} - \frac{1218576838}{10054661351} a^{6} - \frac{1720030428}{10054661351} a^{5} + \frac{532544716}{10054661351} a^{4} + \frac{3241851686}{10054661351} a^{3} - \frac{1913186817}{10054661351} a^{2} + \frac{3417032247}{10054661351} a - \frac{1784913673}{10054661351}$
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: | \( \frac{3504760219}{10054661351} a^{15} - \frac{20878761079}{20109322702} a^{14} - \frac{14259775673}{20109322702} a^{13} + \frac{10277818617}{20109322702} a^{12} + \frac{148408792449}{10054661351} a^{11} - \frac{349505766769}{20109322702} a^{10} - \frac{184322844871}{20109322702} a^{9} - \frac{399611016767}{20109322702} a^{8} + \frac{743419312361}{10054661351} a^{7} - \frac{680905723927}{20109322702} a^{6} + \frac{82869843885}{20109322702} a^{5} - \frac{557570661429}{20109322702} a^{4} + \frac{86641270178}{10054661351} a^{3} + \frac{141069713556}{10054661351} a^{2} - \frac{97198897568}{10054661351} a + \frac{41616916195}{10054661351} \) (order $24$) | 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: | \( 4536.17900444 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times D_4$ (as 16T25):
| A solvable group of order 32 |
| The 20 conjugacy class representatives for $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
Intermediate fields
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 | R | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |