Normalized defining polynomial
\( x^{16} + 52 x^{14} + 1076 x^{12} + 11236 x^{10} + 61786 x^{8} + 168260 x^{6} + 182324 x^{4} + 32660 x^{2} + 529 \)
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: | \(346004614354272376477188096=2^{42}\cdot 3^{12}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $45.57$ | 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: | $2, 3, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{10} a^{8} - \frac{7}{20} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{3}{40} a^{2} - \frac{1}{2} a - \frac{1}{20}$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{11} + \frac{1}{40} a^{9} - \frac{1}{8} a^{8} + \frac{2}{5} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{7}{40} a^{3} - \frac{1}{4} a^{2} + \frac{3}{40} a - \frac{1}{8}$, $\frac{1}{169327323880} a^{14} - \frac{247147943}{169327323880} a^{12} - \frac{19259731}{84663661940} a^{10} + \frac{9633839507}{84663661940} a^{8} - \frac{1}{2} a^{7} - \frac{5721436609}{13025178760} a^{6} - \frac{13369242093}{169327323880} a^{4} - \frac{1}{2} a^{3} - \frac{4370127457}{21165915485} a^{2} + \frac{168118426}{920257195}$, $\frac{1}{169327323880} a^{15} - \frac{247147943}{169327323880} a^{13} - \frac{19259731}{84663661940} a^{11} - \frac{1898236471}{169327323880} a^{9} - \frac{1}{8} a^{8} - \frac{2465141919}{13025178760} a^{7} - \frac{1}{4} a^{6} + \frac{71294419847}{169327323880} a^{5} - \frac{38646425313}{84663661940} a^{3} + \frac{1}{4} a^{2} + \frac{424690213}{7362057560} a + \frac{3}{8}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 147856.675539 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$(C_2^2\times C_4):C_2$ (as 16T37):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $(C_2^2\times C_4):C_2$ |
| Character table for $(C_2^2\times C_4):C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.39744.1, 4.4.158976.2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.404373897216.34, 8.8.101093474304.2, 8.0.44930433024.1 |
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 | R | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| 3 | Data not computed | ||||||
| $23$ | 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |