Normalized defining polynomial
\( x^{16} - 4 x^{15} + 24 x^{13} - 33 x^{12} - 71 x^{11} + 177 x^{10} + 157 x^{9} - 504 x^{8} - 235 x^{7} + 949 x^{6} + 60 x^{5} - 1327 x^{4} + 244 x^{3} + 1217 x^{2} - 133 x - 439 \)
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: | \(505577180252520423441=3^{2}\cdot 19^{6}\cdot 103^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.68$ | 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: | $3, 19, 103$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{130120835182161217} a^{15} - \frac{8134755235075375}{130120835182161217} a^{14} + \frac{60560763688061879}{130120835182161217} a^{13} + \frac{63066419525619332}{130120835182161217} a^{12} - \frac{24531988559923214}{130120835182161217} a^{11} + \frac{30914528681321174}{130120835182161217} a^{10} + \frac{54354550174211476}{130120835182161217} a^{9} + \frac{18315810601121760}{130120835182161217} a^{8} - \frac{39829847870403673}{130120835182161217} a^{7} - \frac{45714999181249694}{130120835182161217} a^{6} + \frac{23000320003398130}{130120835182161217} a^{5} + \frac{62077460766827834}{130120835182161217} a^{4} - \frac{14203644650781386}{130120835182161217} a^{3} + \frac{24051141815741674}{130120835182161217} a^{2} + \frac{17432600358484737}{130120835182161217} a - \frac{23989662712680791}{130120835182161217}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 8619.51829856 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 6144 |
| The 60 conjugacy class representatives for t16n1683 are not computed |
| Character table for t16n1683 is not computed |
Intermediate fields
| 4.4.1957.1, 8.2.11489547.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.8.4.1 | $x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $103$ | 103.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 103.4.2.2 | $x^{4} - 103 x^{2} + 53045$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 103.8.4.1 | $x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |