Normalized defining polynomial
\( x^{16} - 2 x^{14} - 39 x^{13} + 7 x^{12} + 81 x^{11} + 431 x^{10} - 222 x^{9} - 235 x^{8} - 2112 x^{7} - 892 x^{6} + 3633 x^{5} + 3712 x^{4} - 1134 x^{3} - 4417 x^{2} + 588 x + 931 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(214314879732757562500000000=2^{8}\cdot 5^{12}\cdot 7^{8}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.23$ | 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: | $2, 5, 7, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{91} a^{14} - \frac{1}{13} a^{13} - \frac{9}{91} a^{12} - \frac{3}{7} a^{11} - \frac{5}{13} a^{10} - \frac{38}{91} a^{9} + \frac{18}{91} a^{8} - \frac{40}{91} a^{7} + \frac{38}{91} a^{6} + \frac{44}{91} a^{5} + \frac{3}{7} a^{4} - \frac{2}{13} a^{3} - \frac{12}{91} a^{2} + \frac{1}{13} a + \frac{4}{13}$, $\frac{1}{5339231602282697798015701} a^{15} - \frac{20276408449972284384459}{5339231602282697798015701} a^{14} - \frac{1643002587912039128047723}{5339231602282697798015701} a^{13} - \frac{2075725972947838303083571}{5339231602282697798015701} a^{12} + \frac{1029797064586306596063683}{5339231602282697798015701} a^{11} + \frac{2357819271859733134846425}{5339231602282697798015701} a^{10} - \frac{1957454838801946434458479}{5339231602282697798015701} a^{9} + \frac{464906303419013054426960}{5339231602282697798015701} a^{8} + \frac{2415800664813212523822834}{5339231602282697798015701} a^{7} + \frac{985139900900450367157300}{5339231602282697798015701} a^{6} + \frac{339199345342817599360503}{5339231602282697798015701} a^{5} + \frac{2071310035622582018169021}{5339231602282697798015701} a^{4} + \frac{96490022103871815989495}{5339231602282697798015701} a^{3} + \frac{755529628906788515289736}{5339231602282697798015701} a^{2} + \frac{365804066244778188206246}{762747371754671114002243} a + \frac{258883264320718274268946}{762747371754671114002243}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 6581032.58801 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$(C_2^2\times C_4).C_2^4$ (as 16T471):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$ |
| Character table for $(C_2^2\times C_4).C_2^4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.6125.1, 4.4.177625.1, 8.8.31550640625.1 |
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 | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.4 | $x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $29$ | 29.4.2.2 | $x^{4} - 29 x^{2} + 2523$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |