Normalized defining polynomial
\( x^{16} - 7 x^{15} + 21 x^{14} - 30 x^{13} + x^{12} + 84 x^{11} - 145 x^{10} + 89 x^{9} + 42 x^{8} - 77 x^{7} - 85 x^{6} - 3 x^{5} + 121 x^{4} + 30 x^{3} - 66 x^{2} - 26 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(145182542873291015625=3^{6}\cdot 5^{12}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.20$ | 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: | $3, 5, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{21} a^{10} + \frac{1}{3} a^{9} + \frac{4}{21} a^{8} - \frac{10}{21} a^{7} - \frac{3}{7} a^{6} - \frac{10}{21} a^{5} + \frac{3}{7} a^{4} - \frac{1}{21} a^{3} + \frac{5}{21} a^{2} - \frac{8}{21} a + \frac{5}{21}$, $\frac{1}{21} a^{11} - \frac{1}{7} a^{9} + \frac{4}{21} a^{8} - \frac{2}{21} a^{7} - \frac{10}{21} a^{6} - \frac{5}{21} a^{5} - \frac{1}{21} a^{4} - \frac{3}{7} a^{3} - \frac{1}{21} a^{2} - \frac{2}{21} a + \frac{1}{3}$, $\frac{1}{273} a^{12} - \frac{5}{273} a^{11} + \frac{1}{273} a^{10} + \frac{5}{273} a^{9} - \frac{16}{91} a^{8} - \frac{124}{273} a^{7} - \frac{18}{91} a^{6} - \frac{37}{273} a^{5} - \frac{4}{21} a^{4} + \frac{82}{273} a^{3} - \frac{103}{273} a^{2} - \frac{12}{91} a - \frac{40}{91}$, $\frac{1}{273} a^{13} + \frac{2}{273} a^{11} - \frac{1}{91} a^{10} + \frac{27}{91} a^{9} - \frac{1}{7} a^{8} - \frac{50}{273} a^{7} + \frac{32}{91} a^{6} + \frac{12}{91} a^{5} - \frac{16}{91} a^{4} + \frac{86}{273} a^{3} - \frac{32}{91} a^{2} + \frac{25}{273} a + \frac{3}{13}$, $\frac{1}{273} a^{14} - \frac{2}{91} a^{11} + \frac{1}{273} a^{10} - \frac{10}{273} a^{9} - \frac{15}{91} a^{8} + \frac{58}{273} a^{7} - \frac{116}{273} a^{6} + \frac{4}{21} a^{5} + \frac{47}{273} a^{4} - \frac{5}{21} a^{3} + \frac{127}{273} a^{2} - \frac{34}{273} a + \frac{32}{273}$, $\frac{1}{3887247} a^{15} + \frac{4678}{3887247} a^{14} + \frac{2630}{3887247} a^{13} + \frac{55}{44681} a^{12} - \frac{51416}{3887247} a^{11} - \frac{10397}{1295749} a^{10} + \frac{261179}{3887247} a^{9} - \frac{143977}{1295749} a^{8} - \frac{29752}{99673} a^{7} - \frac{137582}{1295749} a^{6} - \frac{1625185}{3887247} a^{5} + \frac{112837}{299019} a^{4} + \frac{250945}{1295749} a^{3} - \frac{1634258}{3887247} a^{2} + \frac{9977}{42717} a + \frac{241057}{3887247}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 7095.08053777 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times D_8$ (as 16T29):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2\times D_8$ |
| Character table for $C_2\times D_8$ |
Intermediate fields
| \(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), 4.2.507.1, 4.2.12675.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.2.12049171875.1, 8.2.12049171875.2, 8.4.160655625.1 |
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 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5 | Data not computed | ||||||
| $13$ | 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 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |