Normalized defining polynomial
\( x^{21} - 66 x^{19} - 44 x^{18} + 1782 x^{17} + 2376 x^{16} - 24669 x^{15} - 50922 x^{14} + 173574 x^{13} + 545848 x^{12} - 410832 x^{11} - 2968128 x^{10} - 1876656 x^{9} + 6614208 x^{8} + 12455691 x^{7} + 2810462 x^{6} - 15172164 x^{5} - 22782312 x^{4} - 16227568 x^{3} - 6574176 x^{2} - 1460928 x - 139136 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[21, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2414563091823149581136287358490706306288929226752=2^{14}\cdot 3^{21}\cdot 11\cdot 1009^{9}\cdot 1087^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $201.35$ | 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: | $2, 3, 11, 1009, 1087$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} + \frac{3}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{13}{32} a^{14} + \frac{1}{8} a^{13} + \frac{3}{32} a^{12} - \frac{3}{16} a^{11} - \frac{5}{64} a^{10} + \frac{3}{16} a^{9} + \frac{3}{32} a^{8} + \frac{5}{16} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{11}{64} a^{2} - \frac{3}{16} a - \frac{5}{16}$, $\frac{1}{512} a^{17} + \frac{11}{256} a^{15} + \frac{37}{128} a^{14} - \frac{37}{256} a^{13} + \frac{21}{64} a^{12} - \frac{173}{512} a^{11} - \frac{117}{256} a^{10} + \frac{87}{256} a^{9} - \frac{15}{64} a^{8} + \frac{1}{32} a^{7} - \frac{7}{16} a^{6} - \frac{9}{32} a^{5} - \frac{3}{8} a^{4} + \frac{139}{512} a^{3} + \frac{47}{256} a^{2} + \frac{33}{128} a - \frac{27}{64}$, $\frac{1}{4096} a^{18} - \frac{1}{2048} a^{17} + \frac{11}{2048} a^{16} + \frac{13}{512} a^{15} - \frac{697}{2048} a^{14} + \frac{335}{1024} a^{13} - \frac{509}{4096} a^{12} - \frac{57}{256} a^{11} + \frac{321}{2048} a^{10} - \frac{501}{1024} a^{9} - \frac{5}{16} a^{8} + \frac{5}{16} a^{7} - \frac{45}{256} a^{6} - \frac{13}{128} a^{5} - \frac{501}{4096} a^{4} - \frac{215}{512} a^{3} - \frac{71}{512} a^{2} - \frac{15}{128} a - \frac{101}{256}$, $\frac{1}{32768} a^{19} - \frac{1}{8192} a^{18} + \frac{13}{16384} a^{17} + \frac{15}{8192} a^{16} - \frac{801}{16384} a^{15} - \frac{255}{1024} a^{14} + \frac{13195}{32768} a^{13} - \frac{6091}{16384} a^{12} - \frac{6959}{16384} a^{11} + \frac{613}{4096} a^{10} - \frac{171}{4096} a^{9} + \frac{31}{128} a^{8} - \frac{973}{2048} a^{7} + \frac{5}{32} a^{6} + \frac{12619}{32768} a^{5} - \frac{2407}{16384} a^{4} + \frac{1383}{4096} a^{3} + \frac{809}{2048} a^{2} - \frac{809}{2048} a + \frac{357}{1024}$, $\frac{1}{262144} a^{20} + \frac{1}{131072} a^{19} + \frac{1}{131072} a^{18} + \frac{27}{32768} a^{17} - \frac{621}{131072} a^{16} + \frac{3749}{65536} a^{15} + \frac{128075}{262144} a^{14} - \frac{16021}{65536} a^{13} + \frac{22031}{131072} a^{12} - \frac{11459}{65536} a^{11} - \frac{12877}{32768} a^{10} - \frac{17}{16384} a^{9} - \frac{2093}{16384} a^{8} + \frac{2361}{8192} a^{7} - \frac{120501}{262144} a^{6} + \frac{9533}{65536} a^{5} - \frac{4455}{65536} a^{4} - \frac{1617}{8192} a^{3} - \frac{2099}{16384} a^{2} + \frac{1013}{4096} a - \frac{977}{4096}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $20$ | 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: | \( 308308678162000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 3919104 |
| The 288 conjugacy class representatives for t21n131 are not computed |
| Character table for t21n131 is not computed |
Intermediate fields
| 7.7.1027243729.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 siblings: | data not computed |
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 | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $21$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.32 | $x^{14} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} - 2 x^{7} + 4 x^{6} - 2 x^{2} + 2 x + 3$ | $2$ | $7$ | $14$ | 14T21 | $[2, 2, 2, 2, 2, 2]^{7}$ | |
| 3 | Data not computed | ||||||
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 1009 | Data not computed | ||||||
| 1087 | Data not computed | ||||||