Normalized defining polynomial
\( x^{16} - 4 x^{15} + 2 x^{14} + 10 x^{13} - 5 x^{12} - 32 x^{11} + 35 x^{10} + 24 x^{9} - 42 x^{8} - 8 x^{7} + 27 x^{6} - 4 x^{5} - 5 x^{4} - 4 x^{3} + 11 x^{2} - 6 x + 1 \)
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: | \(287365367070720000=2^{24}\cdot 3^{8}\cdot 5^{4}\cdot 4177\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.34$ | 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: | $2, 3, 5, 4177$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{14} a^{13} - \frac{1}{7} a^{12} + \frac{3}{14} a^{11} - \frac{3}{7} a^{10} + \frac{1}{14} a^{9} + \frac{1}{7} a^{8} - \frac{3}{14} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{14} a - \frac{2}{7}$, $\frac{1}{14} a^{14} - \frac{1}{14} a^{12} + \frac{3}{14} a^{10} + \frac{2}{7} a^{9} + \frac{1}{14} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{14} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{2702} a^{15} + \frac{4}{1351} a^{14} - \frac{95}{2702} a^{13} + \frac{607}{2702} a^{12} - \frac{55}{2702} a^{11} - \frac{885}{2702} a^{10} - \frac{1321}{2702} a^{9} + \frac{1349}{2702} a^{8} - \frac{419}{1351} a^{7} - \frac{993}{2702} a^{6} + \frac{135}{1351} a^{5} + \frac{653}{1351} a^{4} - \frac{133}{386} a^{3} + \frac{395}{1351} a^{2} - \frac{562}{1351} a - \frac{563}{2702}$
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: | \( \frac{4960}{1351} a^{15} - \frac{18027}{1351} a^{14} + \frac{3194}{1351} a^{13} + \frac{51258}{1351} a^{12} - \frac{6074}{1351} a^{11} - \frac{162321}{1351} a^{10} + \frac{114060}{1351} a^{9} + \frac{165131}{1351} a^{8} - \frac{150186}{1351} a^{7} - \frac{98929}{1351} a^{6} + \frac{100140}{1351} a^{5} + \frac{19594}{1351} a^{4} - \frac{19342}{1351} a^{3} - \frac{27292}{1351} a^{2} + \frac{43962}{1351} a - \frac{11929}{1351} \) (order $6$) | 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: | \( 321.150081014 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 4096 |
| The 88 conjugacy class representatives for t16n1577 are not computed |
| Character table for t16n1577 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 8.0.8294400.1 |
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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 3 | Data not computed | ||||||
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 4177 | Data not computed | ||||||