Normalized defining polynomial
\( x^{16} - 3 x^{14} - 10 x^{13} + 8 x^{12} + 70 x^{11} - 16 x^{10} - 60 x^{9} - 195 x^{8} + 300 x^{7} + 349 x^{6} - 1710 x^{5} + 3053 x^{4} - 1980 x^{3} + 3267 x^{2} - 13310 x + 14641 \)
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: | \(950586000400000000000000=2^{16}\cdot 5^{14}\cdot 29^{2}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.52$ | 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: | $2, 5, 29, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{11} + \frac{1}{11} a^{10} - \frac{3}{11} a^{9} + \frac{4}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} + \frac{3}{11} a^{5} + \frac{3}{11} a^{4} - \frac{3}{11} a^{3} - \frac{5}{11} a^{2} - \frac{5}{11} a$, $\frac{1}{3509} a^{14} - \frac{10}{319} a^{13} - \frac{1455}{3509} a^{12} + \frac{1530}{3509} a^{11} + \frac{1229}{3509} a^{10} - \frac{1657}{3509} a^{9} + \frac{1601}{3509} a^{8} + \frac{853}{3509} a^{7} + \frac{113}{3509} a^{6} + \frac{24}{121} a^{5} - \frac{465}{3509} a^{4} + \frac{314}{3509} a^{3} - \frac{390}{3509} a^{2} - \frac{9}{319} a + \frac{7}{29}$, $\frac{1}{5374093244405091947520439} a^{15} - \frac{9627780808227062204}{488553931309553813410949} a^{14} - \frac{195424268481604727702833}{5374093244405091947520439} a^{13} + \frac{197968962454952142732873}{5374093244405091947520439} a^{12} + \frac{1337959295021453555144757}{5374093244405091947520439} a^{11} + \frac{40989675584229481875616}{5374093244405091947520439} a^{10} - \frac{837625368349438321090718}{5374093244405091947520439} a^{9} - \frac{176413653190872587557325}{5374093244405091947520439} a^{8} - \frac{2331398785874320515464449}{5374093244405091947520439} a^{7} + \frac{2229510349631510002164990}{5374093244405091947520439} a^{6} - \frac{2393836687947919372843809}{5374093244405091947520439} a^{5} - \frac{2119798117817317301153663}{5374093244405091947520439} a^{4} - \frac{933697333209571765440974}{5374093244405091947520439} a^{3} + \frac{197007559920193904063531}{488553931309553813410949} a^{2} + \frac{426894632535393172335}{44413993755413983037359} a + \frac{1413198066054068464368}{4037635795946725730669}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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: | \( 23258.903495 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^5.C_2$ (as 16T610):
| A solvable group of order 256 |
| The 40 conjugacy class representatives for $C_2^2.C_2^5.C_2$ |
| Character table for $C_2^2.C_2^5.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.5125.1, 4.4.16400.1, 8.8.6724000000.1 |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $5$ | 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} - 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| $29$ | 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.2 | $x^{4} - 29 x^{2} + 2523$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |