Normalized defining polynomial
\( x^{16} - x^{15} + 3 x^{14} + 8 x^{13} + 14 x^{12} + 25 x^{11} + 54 x^{10} + 28 x^{9} + 77 x^{8} - 28 x^{7} + 54 x^{6} - 25 x^{5} + 14 x^{4} - 8 x^{3} + 3 x^{2} + x + 1 \)
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: | \(114713585069001347360475625=5^{4}\cdot 17^{12}\cdot 17749^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.53$ | 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: | $5, 17, 17749$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5}$, $\frac{1}{50} a^{12} - \frac{2}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{50} a^{9} - \frac{2}{5} a^{8} - \frac{2}{25} a^{7} - \frac{3}{50} a^{6} + \frac{2}{25} a^{5} - \frac{2}{5} a^{4} + \frac{1}{50} a^{3} - \frac{2}{25} a^{2} + \frac{2}{25} a + \frac{1}{50}$, $\frac{1}{50} a^{13} + \frac{3}{50} a^{10} + \frac{8}{25} a^{9} - \frac{7}{25} a^{8} + \frac{11}{50} a^{7} + \frac{6}{25} a^{6} + \frac{8}{25} a^{5} + \frac{1}{50} a^{4} + \frac{2}{5} a^{3} + \frac{4}{25} a^{2} + \frac{7}{50} a + \frac{12}{25}$, $\frac{1}{250} a^{14} - \frac{1}{250} a^{12} + \frac{7}{250} a^{11} + \frac{97}{250} a^{9} + \frac{1}{250} a^{8} - \frac{42}{125} a^{7} - \frac{11}{250} a^{6} - \frac{103}{250} a^{5} + \frac{7}{25} a^{4} + \frac{107}{250} a^{3} + \frac{91}{250} a^{2} - \frac{12}{25} a + \frac{119}{250}$, $\frac{1}{250} a^{15} - \frac{1}{250} a^{13} + \frac{1}{125} a^{12} + \frac{2}{25} a^{11} + \frac{17}{250} a^{10} + \frac{28}{125} a^{9} + \frac{58}{125} a^{8} + \frac{9}{250} a^{7} + \frac{6}{125} a^{6} + \frac{1}{5} a^{5} + \frac{107}{250} a^{4} + \frac{43}{125} a^{3} + \frac{1}{5} a^{2} - \frac{101}{250} a + \frac{19}{50}$
Class group and class number
$C_{15}$, which has order $15$ (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: | \( 195050.432151 \) (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 t16n1385 are not computed |
| Character table for t16n1385 is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 8.0.428417712181.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.2 | $x^{8} + 25 x^{4} - 250 x^{2} + 1250$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $17$ | 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 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17749 | Data not computed | ||||||