Normalized defining polynomial
\( x^{16} - 28 x^{14} - 122 x^{12} + 3224 x^{10} + 4176 x^{8} - 71995 x^{6} - 129645 x^{4} + 207088 x^{2} + 318028 \)
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: | \(9167390055669127803424473088=2^{20}\cdot 43^{5}\cdot 2777^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.93$ | 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: | $2, 43, 2777$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{43} a^{12} + \frac{15}{43} a^{10} + \frac{7}{43} a^{8} - \frac{1}{43} a^{6} + \frac{5}{43} a^{4} - \frac{13}{43} a^{2}$, $\frac{1}{43} a^{13} + \frac{15}{43} a^{11} + \frac{7}{43} a^{9} - \frac{1}{43} a^{7} + \frac{5}{43} a^{5} - \frac{13}{43} a^{3}$, $\frac{1}{120891335147577504428} a^{14} - \frac{295586407096390103}{60445667573788752214} a^{12} - \frac{10906618032853746727}{60445667573788752214} a^{10} - \frac{12779659385759919794}{30222833786894376107} a^{8} + \frac{405593163812718514}{30222833786894376107} a^{6} + \frac{885967178582803475}{5256145006416413236} a^{4} - \frac{295078061149366113}{2811426398780872196} a^{2} + \frac{6842570978712225}{32691004636986886}$, $\frac{1}{241782670295155008856} a^{15} + \frac{1110126792294045995}{120891335147577504428} a^{13} - \frac{50266587615785957471}{120891335147577504428} a^{11} - \frac{7859663187893393451}{60445667573788752214} a^{9} + \frac{14962785175505938286}{30222833786894376107} a^{7} - \frac{3758998175924724501}{10512290012832826472} a^{5} + \frac{1666382217069847047}{5622852797561744392} a^{3} + \frac{6842570978712225}{65382009273973772} a$
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: | \( 151905574.715 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 6144 |
| The 63 conjugacy class representatives for t16n1666 are not computed |
| Character table for t16n1666 is not computed |
Intermediate fields
| 4.4.2777.1, 8.8.1326417388.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.8.6 | $x^{4} + 6 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $D_{4}$ | $[2, 3]^{2}$ |
| 2.12.12.5 | $x^{12} + 52 x^{10} - 11 x^{8} - 8 x^{6} - 45 x^{4} - 44 x^{2} - 9$ | $2$ | $6$ | $12$ | 12T51 | $[2, 2, 2, 2]^{6}$ | |
| $43$ | 43.3.0.1 | $x^{3} - x + 10$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 43.3.0.1 | $x^{3} - x + 10$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.6.3.1 | $x^{6} - 86 x^{4} + 1849 x^{2} - 7950700$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 2777 | Data not computed | ||||||