Normalized defining polynomial
\( x^{16} + 29 x^{14} + 208 x^{12} - 582 x^{10} - 11202 x^{8} - 37656 x^{6} - 43758 x^{4} - 11934 x^{2} + 1377 \)
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: | \(134425712786353297191406374912=2^{12}\cdot 3^{8}\cdot 17^{9}\cdot 59^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $66.15$ | 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: | $2, 3, 17, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{9} - \frac{4}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{27} a^{12} + \frac{1}{27} a^{10} - \frac{1}{9} a^{6}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{9} - \frac{1}{9} a^{7} + \frac{4}{9} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{675203251518} a^{14} - \frac{1}{54} a^{13} - \frac{141610631}{225067750506} a^{12} + \frac{7905229229}{675203251518} a^{10} + \frac{1}{54} a^{9} - \frac{4067665030}{112533875253} a^{8} + \frac{1}{18} a^{7} + \frac{4653973823}{75022583502} a^{6} - \frac{2}{9} a^{5} - \frac{12599562101}{75022583502} a^{4} - \frac{1}{6} a^{3} + \frac{12789913415}{37511291751} a^{2} - \frac{1}{2} a + \frac{5998377717}{25007527834}$, $\frac{1}{675203251518} a^{15} + \frac{6039466012}{337601625759} a^{13} - \frac{1}{54} a^{12} + \frac{7905229229}{675203251518} a^{11} + \frac{1}{27} a^{10} - \frac{36909754097}{675203251518} a^{9} - \frac{1}{18} a^{8} + \frac{729078776}{112533875253} a^{7} - \frac{1}{18} a^{6} + \frac{12216369365}{225067750506} a^{5} + \frac{1}{3} a^{4} - \frac{12312997585}{25007527834} a^{3} - \frac{1}{6} a^{2} - \frac{3252693100}{12503763917} a - \frac{1}{2}$
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: | \( 425732472.715 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 49152 |
| The 104 conjugacy class representatives for t16n1849 are not computed |
| Character table for t16n1849 is not computed |
Intermediate fields
| 4.4.51153.1, 8.8.463143405393.1 |
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 | $16$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | $16$ | $16$ | $16$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | R |
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 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.12.12.19 | $x^{12} - 6 x^{10} + 27 x^{8} - 4 x^{6} + 7 x^{4} + 10 x^{2} + 29$ | $2$ | $6$ | $12$ | 12T105 | $[2, 2, 2, 2]^{12}$ | |
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.6.5.1 | $x^{6} - 17$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 17.6.4.1 | $x^{6} + 136 x^{3} + 7803$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $59$ | 59.4.0.1 | $x^{4} - x + 14$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 59.4.0.1 | $x^{4} - x + 14$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.4.3.1 | $x^{4} + 177$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 59.4.3.1 | $x^{4} + 177$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |