Normalized defining polynomial
\( x^{16} - 8 x^{15} + 31 x^{14} - 77 x^{13} + 173 x^{12} - 371 x^{11} + 621 x^{10} - 836 x^{9} + 1463 x^{8} - 1828 x^{7} + 1237 x^{6} - 1805 x^{5} + 2223 x^{4} + 1361 x^{3} + 2339 x^{2} + 726 x + 277 \)
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: | \(166625271956783950277393=17^{7}\cdot 67^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.27$ | 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: | $17, 67$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} - \frac{4}{17} a^{12} - \frac{4}{17} a^{11} + \frac{8}{17} a^{10} - \frac{6}{17} a^{9} - \frac{1}{17} a^{8} - \frac{4}{17} a^{7} + \frac{3}{17} a^{6} - \frac{4}{17} a^{5} + \frac{5}{17} a^{4} - \frac{3}{17} a^{3} - \frac{6}{17} a^{2} + \frac{1}{17} a - \frac{7}{17}$, $\frac{1}{323} a^{14} + \frac{6}{323} a^{13} - \frac{61}{323} a^{12} - \frac{49}{323} a^{11} - \frac{113}{323} a^{10} - \frac{112}{323} a^{9} - \frac{116}{323} a^{8} + \frac{7}{17} a^{7} - \frac{161}{323} a^{6} - \frac{35}{323} a^{5} + \frac{13}{323} a^{4} + \frac{15}{323} a^{3} + \frac{145}{323} a^{2} + \frac{105}{323} a - \frac{70}{323}$, $\frac{1}{3336976091460318904097} a^{15} + \frac{703539701938356803}{3336976091460318904097} a^{14} - \frac{218468017298506374}{12984342768328089121} a^{13} - \frac{924092863624701858674}{3336976091460318904097} a^{12} + \frac{799247029246860900934}{3336976091460318904097} a^{11} - \frac{44179285712909554893}{3336976091460318904097} a^{10} + \frac{1486940118194324040434}{3336976091460318904097} a^{9} - \frac{766255704249583510655}{3336976091460318904097} a^{8} + \frac{17242200236210506450}{46999663260004491607} a^{7} - \frac{389973387927142674223}{3336976091460318904097} a^{6} + \frac{1221558122655801767673}{3336976091460318904097} a^{5} - \frac{841493778943863961322}{3336976091460318904097} a^{4} + \frac{1347250039163416884464}{3336976091460318904097} a^{3} - \frac{1502484186372710577074}{3336976091460318904097} a^{2} - \frac{1562390083768834835658}{3336976091460318904097} a - \frac{1351514149316324393195}{3336976091460318904097}$
Class group and class number
$C_{5}$, which has order $5$ (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: | \( 26320.2365193 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4.D_4:C_4$ (as 16T289):
| A solvable group of order 128 |
| The 44 conjugacy class representatives for $C_4.D_4:C_4$ |
| Character table for $C_4.D_4:C_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-67}) \), 4.0.76313.1, 8.0.99002457473.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | $16$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | $16$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $17$ | $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.8.7.3 | $x^{8} - 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
| $67$ | 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |