Normalized defining polynomial
\( x^{16} - 3 x^{15} - x^{14} + 4 x^{13} + 6 x^{12} - 15 x^{11} + 5 x^{10} + 38 x^{9} - 21 x^{8} - 19 x^{7} - 45 x^{6} + 42 x^{5} - 72 x^{4} + 162 x^{3} - 30 x^{2} + 64 x - 13 \)
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: | \(1491512927308188160000=2^{12}\cdot 5^{4}\cdot 17^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.05$ | 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: | $2, 5, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{10} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{119} a^{14} + \frac{8}{119} a^{13} - \frac{1}{17} a^{12} + \frac{25}{119} a^{11} + \frac{4}{119} a^{10} - \frac{9}{119} a^{9} - \frac{11}{119} a^{8} - \frac{19}{119} a^{7} - \frac{29}{119} a^{6} + \frac{54}{119} a^{5} - \frac{57}{119} a^{4} - \frac{3}{7} a^{3} - \frac{1}{119} a^{2} - \frac{2}{119} a + \frac{8}{119}$, $\frac{1}{638624626381} a^{15} + \frac{14277964}{638624626381} a^{14} - \frac{21954801322}{638624626381} a^{13} - \frac{38088161205}{638624626381} a^{12} - \frac{128588831787}{638624626381} a^{11} + \frac{31055313389}{638624626381} a^{10} - \frac{196785785795}{638624626381} a^{9} + \frac{8499792266}{91232089483} a^{8} - \frac{54675736468}{638624626381} a^{7} - \frac{16767535488}{37566154493} a^{6} + \frac{89530662026}{638624626381} a^{5} - \frac{309754000513}{638624626381} a^{4} + \frac{110698191949}{638624626381} a^{3} - \frac{344520732}{91232089483} a^{2} - \frac{139589435659}{638624626381} a - \frac{182633908120}{638624626381}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 27849.4309237 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2048 |
| The 71 conjugacy class representatives for t16n1385 are not computed |
| Character table for t16n1385 is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 8.2.386201104.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.12.15 | $x^{8} + 2 x^{7} + 2 x^{4} + 12$ | $4$ | $2$ | $12$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
| $5$ | 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 5.8.4.2 | $x^{8} + 25 x^{4} - 250 x^{2} + 1250$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.4.3.1 | $x^{4} - 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 17.4.3.1 | $x^{4} - 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |