Normalized defining polynomial
\( x^{16} + 34 x^{14} + 487 x^{12} + 3862 x^{10} + 19025 x^{8} + 62768 x^{6} + 140512 x^{4} + 194816 x^{2} + 194816 \)
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: | \(462119062470656000000000000=2^{32}\cdot 5^{12}\cdot 761^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $46.40$ | 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: | $2, 5, 761$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{20} a^{8} - \frac{1}{4} a^{7} + \frac{1}{10} a^{6} - \frac{1}{4} a^{5} + \frac{1}{5} a^{4} + \frac{1}{4} a^{3} + \frac{3}{20} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{9} - \frac{3}{20} a^{7} - \frac{1}{4} a^{6} - \frac{1}{20} a^{5} - \frac{1}{4} a^{4} + \frac{2}{5} a^{3} + \frac{1}{4} a^{2} - \frac{1}{5} a$, $\frac{1}{80} a^{10} - \frac{1}{40} a^{8} - \frac{9}{80} a^{6} - \frac{19}{40} a^{4} + \frac{9}{80} a^{2} + \frac{9}{20}$, $\frac{1}{160} a^{11} - \frac{1}{80} a^{9} - \frac{9}{160} a^{7} - \frac{1}{4} a^{6} - \frac{19}{80} a^{5} - \frac{1}{4} a^{4} + \frac{9}{160} a^{3} + \frac{1}{4} a^{2} + \frac{9}{40} a$, $\frac{1}{79360} a^{12} + \frac{1}{7936} a^{10} - \frac{69}{15872} a^{8} - \frac{965}{7936} a^{6} + \frac{3677}{15872} a^{4} - \frac{2809}{9920} a^{2} - \frac{79}{992}$, $\frac{1}{317440} a^{13} - \frac{1}{158720} a^{12} - \frac{491}{158720} a^{11} + \frac{491}{79360} a^{10} + \frac{1639}{317440} a^{9} - \frac{1639}{158720} a^{8} - \frac{20201}{158720} a^{7} + \frac{361}{79360} a^{6} + \frac{16401}{317440} a^{5} - \frac{56081}{158720} a^{4} - \frac{3761}{7936} a^{3} + \frac{785}{3968} a^{2} + \frac{7293}{19840} a + \frac{2627}{9920}$, $\frac{1}{1269760} a^{14} - \frac{1}{634880} a^{12} + \frac{241}{40960} a^{10} - \frac{551}{126976} a^{8} - \frac{1}{4} a^{7} - \frac{52523}{253952} a^{6} - \frac{1}{4} a^{5} - \frac{11173}{317440} a^{4} + \frac{1}{4} a^{3} - \frac{23221}{79360} a^{2} + \frac{7137}{19840}$, $\frac{1}{5079040} a^{15} - \frac{1}{2539520} a^{14} - \frac{1}{2539520} a^{13} + \frac{1}{1269760} a^{12} + \frac{241}{163840} a^{11} - \frac{241}{81920} a^{10} + \frac{28989}{2539520} a^{9} - \frac{28989}{1269760} a^{8} + \frac{1134121}{5079040} a^{7} + \frac{135639}{2539520} a^{6} - \frac{53025}{253952} a^{5} - \frac{10463}{126976} a^{4} + \frac{147403}{317440} a^{3} - \frac{68043}{158720} a^{2} - \frac{16671}{79360} a + \frac{16671}{39680}$
Class group and class number
$C_{2}\times C_{8}$, which has order $16$ (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: | \( 439006.410817 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2048 |
| The 71 conjugacy class representatives for t16n1398 are not computed |
| Character table for t16n1398 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.48704000000.2 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.3 | $x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 761 | Data not computed | ||||||