Normalized defining polynomial
\( x^{16} - 4 x^{15} + 8 x^{14} - 23 x^{13} + 140 x^{12} - 406 x^{11} + 671 x^{10} - 824 x^{9} + 1122 x^{8} - 1181 x^{7} + 578 x^{6} - 34 x^{5} + 151 x^{4} + 72 x^{3} + 208 x^{2} + 104 x + 16 \)
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: | \(189537365823025172265625=5^{8}\cdot 13^{8}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.50$ | 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: | $5, 13, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{581} a^{12} + \frac{163}{581} a^{11} - \frac{286}{581} a^{10} - \frac{39}{581} a^{9} - \frac{191}{581} a^{8} + \frac{41}{83} a^{7} - \frac{11}{581} a^{6} - \frac{104}{581} a^{5} - \frac{277}{581} a^{4} + \frac{19}{83} a^{3} - \frac{108}{581} a^{2} + \frac{180}{581} a - \frac{50}{581}$, $\frac{1}{1162} a^{13} + \frac{226}{581} a^{11} + \frac{99}{1162} a^{10} + \frac{178}{581} a^{9} + \frac{23}{581} a^{8} + \frac{269}{1162} a^{7} - \frac{27}{581} a^{6} - \frac{87}{581} a^{5} + \frac{547}{1162} a^{4} - \frac{145}{581} a^{3} + \frac{177}{581} a^{2} + \frac{241}{1162} a + \frac{8}{581}$, $\frac{1}{6972} a^{14} - \frac{53}{996} a^{11} - \frac{43}{581} a^{10} - \frac{153}{1162} a^{9} + \frac{1775}{6972} a^{8} - \frac{260}{581} a^{7} + \frac{1237}{3486} a^{6} + \frac{1133}{2324} a^{5} - \frac{1453}{3486} a^{4} - \frac{277}{1162} a^{3} + \frac{1415}{6972} a^{2} - \frac{1}{1743} a - \frac{160}{1743}$, $\frac{1}{49220829679224} a^{15} - \frac{288906749}{24610414839612} a^{14} + \frac{184703006}{2050867903301} a^{13} + \frac{16747314889}{49220829679224} a^{12} - \frac{170332055471}{3515773548516} a^{11} - \frac{324353923521}{8203471613204} a^{10} - \frac{14791603129621}{49220829679224} a^{9} - \frac{1387658627911}{3515773548516} a^{8} + \frac{294408338119}{3515773548516} a^{7} + \frac{12546742573927}{49220829679224} a^{6} - \frac{4119036977}{74127755541} a^{5} - \frac{6540230153279}{24610414839612} a^{4} + \frac{2552812735061}{7031547097032} a^{3} + \frac{1336096299805}{8203471613204} a^{2} + \frac{784762669298}{2050867903301} a + \frac{391796642537}{878943387129}$
Class group and class number
$C_{4}$, which has order $4$ (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: | \( 25449.0217415 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\wr C_2$ (as 16T46):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
| Character table for $C_2^2\wr C_2$ |
Intermediate fields
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 siblings: | 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |