Normalized defining polynomial
\( x^{16} - 4 x^{15} - 3 x^{14} + 13 x^{13} + 66 x^{12} - 91 x^{11} - 108 x^{10} - 146 x^{9} + 802 x^{8} + 550 x^{7} + 2123 x^{6} + 102 x^{5} + 3277 x^{4} + 2194 x^{3} + 7204 x^{2} + 3820 x + 3851 \)
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: | \(86380562306022715087890625=5^{12}\cdot 29^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $41.79$ | 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: | $5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{238191849129127840260332135} a^{15} - \frac{20462769467600873456080663}{238191849129127840260332135} a^{14} + \frac{2535258607554172997613616}{238191849129127840260332135} a^{13} - \frac{3712088701352259037074871}{47638369825825568052066427} a^{12} - \frac{936308579732241553117099}{238191849129127840260332135} a^{11} - \frac{116083185431510584801843864}{238191849129127840260332135} a^{10} + \frac{67656631272363009497339126}{238191849129127840260332135} a^{9} + \frac{57570831686938697875560781}{238191849129127840260332135} a^{8} - \frac{7034192323788438084163631}{238191849129127840260332135} a^{7} - \frac{38856139178564960247251622}{238191849129127840260332135} a^{6} + \frac{5701882311247493765008248}{18322449933009833866179395} a^{5} + \frac{742499723311863340829614}{238191849129127840260332135} a^{4} - \frac{7388915002195795920494303}{47638369825825568052066427} a^{3} - \frac{13750071718565015448522631}{238191849129127840260332135} a^{2} + \frac{68653610685114760446968187}{238191849129127840260332135} a - \frac{107290700777674168870599726}{238191849129127840260332135}$
Class group and class number
$C_{2}\times C_{12}$, which has order $24$ (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: | \( 82042.9398575 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4\wr C_2$ (as 16T28):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.0.609725.2, 4.4.4205.1, 4.0.121945.1, 8.0.371764575625.4 |
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 8 siblings: | data not computed |
| Degree 16 sibling: | 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.4 | $x^{4} + 40$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 29 | Data not computed | ||||||