Normalized defining polynomial
\( x^{16} - x^{14} - 10 x^{13} + 10 x^{11} + 68 x^{10} + 56 x^{9} + 57 x^{8} - 60 x^{7} - 66 x^{6} - 74 x^{5} + 38 x^{4} + 20 x^{3} + 15 x^{2} - 2 x + 1 \)
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: | \(2706398756904960000=2^{16}\cdot 3^{4}\cdot 5^{4}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.19$ | 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: | $2, 3, 5, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{90} a^{14} + \frac{1}{5} a^{13} - \frac{2}{9} a^{12} + \frac{13}{45} a^{11} - \frac{7}{90} a^{10} - \frac{17}{45} a^{9} - \frac{11}{30} a^{8} - \frac{2}{5} a^{7} - \frac{3}{10} a^{6} + \frac{2}{15} a^{5} + \frac{1}{15} a^{4} - \frac{16}{45} a^{3} - \frac{31}{90} a^{2} - \frac{1}{45} a - \frac{4}{45}$, $\frac{1}{309319150590} a^{15} - \frac{182603687}{34368794510} a^{14} - \frac{1896386239}{154659575295} a^{13} - \frac{33957265667}{154659575295} a^{12} - \frac{57115945393}{309319150590} a^{11} - \frac{104271273187}{309319150590} a^{10} + \frac{45473813497}{103106383530} a^{9} - \frac{9906206687}{34368794510} a^{8} + \frac{10506333171}{34368794510} a^{7} + \frac{47103960343}{103106383530} a^{6} - \frac{20722273541}{51553191765} a^{5} + \frac{10975811951}{154659575295} a^{4} - \frac{126315001849}{309319150590} a^{3} - \frac{38230524911}{309319150590} a^{2} + \frac{45221130212}{154659575295} a + \frac{1488798966}{17184397255}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 376.867694277 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^4$ (as 16T573):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^4.C_2^4$ |
| Character table for $C_2^4.C_2^4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.2.2704.1, 4.4.8112.1, 4.2.507.1, 8.0.1645113600.2, 8.0.182790400.1, 8.4.65804544.1 |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.2 | $x^{8} + 2 x^{7} + 8 x^{2} + 48$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
| 2.8.8.2 | $x^{8} + 2 x^{7} + 8 x^{2} + 48$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $13$ | 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |