Normalized defining polynomial
\( x^{16} - 6 x^{15} + x^{14} + 48 x^{13} - 63 x^{12} - 136 x^{11} + 232 x^{10} + 463 x^{9} - 369 x^{8} - 877 x^{7} + 1637 x^{6} + 2724 x^{5} - 703 x^{4} - 2902 x^{3} - 1779 x^{2} - 411 x - 29 \)
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: | \(1947839620325062500000000=2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 41^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.97$ | 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: | $2, 3, 5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} - \frac{1}{2} a^{11} + \frac{2}{5} a^{9} - \frac{1}{2} a^{7} + \frac{1}{10} a^{6} - \frac{1}{2} a^{4} - \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{2} a^{11} + \frac{2}{5} a^{10} - \frac{1}{2} a^{8} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{1230} a^{14} - \frac{29}{615} a^{13} + \frac{2}{615} a^{12} + \frac{183}{410} a^{11} - \frac{221}{615} a^{10} - \frac{103}{410} a^{9} - \frac{17}{615} a^{8} - \frac{284}{615} a^{7} + \frac{307}{615} a^{6} + \frac{122}{615} a^{5} + \frac{203}{1230} a^{4} + \frac{177}{410} a^{3} + \frac{87}{410} a^{2} + \frac{106}{615} a - \frac{311}{1230}$, $\frac{1}{7372944057934658850} a^{15} - \frac{1189491822149957}{7372944057934658850} a^{14} - \frac{10614745264143877}{7372944057934658850} a^{13} + \frac{547321225319287}{147458881158693177} a^{12} - \frac{2740380834521398723}{7372944057934658850} a^{11} - \frac{3548481691967382953}{7372944057934658850} a^{10} - \frac{356546590804629323}{1474588811586931770} a^{9} + \frac{740320493944473871}{2457648019311552950} a^{8} - \frac{1116356756202850489}{2457648019311552950} a^{7} + \frac{101103332202040754}{737294405793465885} a^{6} - \frac{1049340874010723869}{3686472028967329425} a^{5} - \frac{1088374287533444153}{7372944057934658850} a^{4} + \frac{29647204148140177}{98305920772462118} a^{3} + \frac{80502039262740079}{3686472028967329425} a^{2} + \frac{1603024076392508239}{3686472028967329425} a - \frac{3543062885336957389}{7372944057934658850}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 1802620.16243 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_2^4.C_2$ (as 16T467):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$ |
| Character table for $C_2^3.C_2^4.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.9225.1, 4.4.5125.1, 8.8.2127515625.1 |
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 | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $3$ | 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 41.4.2.2 | $x^{4} - 41 x^{2} + 20172$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 41.8.4.1 | $x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |