Normalized defining polynomial
\( x^{16} - x^{15} - 29 x^{14} + 63 x^{13} + 506 x^{12} + 635 x^{11} + 2254 x^{10} + 4744 x^{9} - 10132 x^{8} - 14206 x^{7} + 2145 x^{6} - 10483 x^{5} - 377 x^{4} + 12557 x^{3} + 1253 x^{2} + 33522 x + 15353 \)
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: | \(108951986979034819834970703125=5^{10}\cdot 101^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.29$ | 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: | $5, 101$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{6}{13} a^{13} - \frac{2}{13} a^{12} + \frac{5}{13} a^{10} - \frac{1}{13} a^{9} - \frac{5}{13} a^{8} + \frac{1}{13} a^{7} + \frac{5}{13} a^{6} + \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{1}{13} a^{3} + \frac{3}{13} a^{2} - \frac{6}{13} a$, $\frac{1}{14908027749253777545074060229857701981963} a^{15} + \frac{491598183161156685619885057593537675361}{14908027749253777545074060229857701981963} a^{14} + \frac{5723221410372813610792645012967236049464}{14908027749253777545074060229857701981963} a^{13} + \frac{1104223948857350484018763070350000968331}{14908027749253777545074060229857701981963} a^{12} + \frac{4813510443967564301021340273775574494916}{14908027749253777545074060229857701981963} a^{11} + \frac{3787148753497633908430719973093445280998}{14908027749253777545074060229857701981963} a^{10} - \frac{7125677621488844507924161550349774574954}{14908027749253777545074060229857701981963} a^{9} - \frac{129831413865404852203259882772472429671}{346698319750087849885443261159481441441} a^{8} - \frac{443363932404653226601852841152971157156}{14908027749253777545074060229857701981963} a^{7} + \frac{1012430600986874554106485660803911802346}{14908027749253777545074060229857701981963} a^{6} + \frac{4877149799088654574426203862058914624797}{14908027749253777545074060229857701981963} a^{5} - \frac{4362902608583365459285542909344248764337}{14908027749253777545074060229857701981963} a^{4} - \frac{7275516347427443399422593352801913756917}{14908027749253777545074060229857701981963} a^{3} - \frac{2809689345695461325597546858043218372377}{14908027749253777545074060229857701981963} a^{2} + \frac{6326913375663262592266809936568939291151}{14908027749253777545074060229857701981963} a - \frac{458744990909676224671320639950532589387}{1146771365327213657313389248450592460151}$
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: | \( 187973620.235 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4:D_4.D_4$ (as 16T681):
| A solvable group of order 256 |
| The 19 conjugacy class representatives for $C_4:D_4.D_4$ |
| Character table for $C_4:D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{101}) \), 4.4.51005.1, 8.4.6568812813125.1 |
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 | $16$ | $16$ | R | $16$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | $16$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.8.7.1 | $x^{8} - 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| 101 | Data not computed | ||||||