Normalized defining polynomial
\( x^{16} + 38 x^{14} + 563 x^{12} + 4248 x^{10} + 17821 x^{8} + 42480 x^{6} + 56300 x^{4} + 38000 x^{2} + 10000 \)
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: | \(53586372784036249600000000=2^{24}\cdot 5^{8}\cdot 19^{4}\cdot 89^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.56$ | 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: | $2, 5, 19, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{30} a^{10} + \frac{4}{15} a^{8} - \frac{7}{30} a^{6} + \frac{4}{15} a^{4} + \frac{11}{30} a^{2} + \frac{1}{3}$, $\frac{1}{30} a^{11} + \frac{4}{15} a^{9} - \frac{7}{30} a^{7} + \frac{4}{15} a^{5} + \frac{11}{30} a^{3} + \frac{1}{3} a$, $\frac{1}{1800} a^{12} - \frac{1}{60} a^{11} - \frac{1}{150} a^{10} + \frac{11}{30} a^{9} - \frac{137}{1800} a^{8} - \frac{23}{60} a^{7} - \frac{151}{900} a^{6} - \frac{2}{15} a^{5} + \frac{421}{1800} a^{4} + \frac{19}{60} a^{3} + \frac{1}{60} a^{2} + \frac{1}{3} a + \frac{7}{18}$, $\frac{1}{18000} a^{13} - \frac{7}{500} a^{11} - \frac{1}{60} a^{10} + \frac{1543}{18000} a^{9} - \frac{2}{15} a^{8} - \frac{2011}{9000} a^{7} - \frac{23}{60} a^{6} + \frac{3901}{18000} a^{5} + \frac{11}{30} a^{4} - \frac{29}{200} a^{3} + \frac{19}{60} a^{2} - \frac{89}{180} a - \frac{1}{6}$, $\frac{1}{54000} a^{14} - \frac{1}{27000} a^{12} - \frac{257}{54000} a^{10} - \frac{1}{2} a^{9} + \frac{1583}{3375} a^{8} - \frac{7999}{54000} a^{6} - \frac{1}{2} a^{5} - \frac{172}{675} a^{4} + \frac{59}{270} a^{2} - \frac{1}{2} a + \frac{25}{54}$, $\frac{1}{540000} a^{15} - \frac{1}{270000} a^{13} + \frac{1543}{540000} a^{11} - \frac{1}{60} a^{10} - \frac{446}{16875} a^{9} + \frac{11}{30} a^{8} - \frac{182599}{540000} a^{7} - \frac{23}{60} a^{6} + \frac{1354}{3375} a^{5} - \frac{2}{15} a^{4} - \frac{298}{675} a^{3} + \frac{19}{60} a^{2} - \frac{13}{108} a + \frac{1}{3}$
Class group and class number
$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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: | \( 59222.6422927 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times D_4^2.C_2$ (as 16T602):
| A solvable group of order 256 |
| The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$ |
| Character table for $C_2\times D_4^2.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.7600.1, 4.4.2225.1, 4.4.676400.2, 8.0.96319360000.1, 8.0.6019960000.1, 8.8.457516960000.2 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.2 | $x^{8} + 2 x^{7} + 8 x^{2} + 48$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
| 2.8.16.12 | $x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 12$ | $4$ | $2$ | $16$ | $C_2^2:C_4$ | $[2, 2, 3]^{2}$ | |
| $5$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $89$ | 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |