Normalized defining polynomial
\( x^{21} - 24 x^{19} - 16 x^{18} + 135 x^{17} + 180 x^{16} + 519 x^{15} + 918 x^{14} - 4653 x^{13} - 13904 x^{12} - 7236 x^{11} + 16440 x^{10} + 49612 x^{9} + 101808 x^{8} + 105621 x^{7} - 51646 x^{6} - 286524 x^{5} - 369432 x^{4} - 255248 x^{3} - 102816 x^{2} - 22848 x - 2176 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[13, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(10423848562958827406082788615011739523268608=2^{14}\cdot 3^{21}\cdot 17^{2}\cdot 29^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $111.81$ | 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: | $2, 3, 17, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{8} a^{11} + \frac{3}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{1}{16} a^{14} - \frac{1}{8} a^{13} - \frac{25}{64} a^{12} + \frac{1}{32} a^{11} - \frac{13}{64} a^{10} + \frac{7}{16} a^{9} + \frac{3}{64} a^{8} + \frac{3}{32} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} - \frac{5}{16} a^{4} + \frac{1}{8} a^{3} + \frac{5}{64} a^{2} + \frac{3}{16} a + \frac{5}{16}$, $\frac{1}{512} a^{17} - \frac{5}{32} a^{14} + \frac{183}{512} a^{13} - \frac{51}{128} a^{12} - \frac{209}{512} a^{11} - \frac{37}{256} a^{10} - \frac{117}{512} a^{9} + \frac{3}{8} a^{8} + \frac{5}{128} a^{7} - \frac{13}{64} a^{6} - \frac{17}{128} a^{5} + \frac{15}{32} a^{4} - \frac{11}{512} a^{3} + \frac{33}{256} a^{2} + \frac{63}{128} a + \frac{27}{64}$, $\frac{1}{69632} a^{18} - \frac{1}{2048} a^{17} + \frac{3}{1088} a^{16} - \frac{77}{4352} a^{15} + \frac{1623}{69632} a^{14} + \frac{12659}{34816} a^{13} + \frac{15239}{69632} a^{12} - \frac{2849}{8704} a^{11} + \frac{14239}{69632} a^{10} - \frac{13915}{34816} a^{9} - \frac{5067}{17408} a^{8} - \frac{1769}{4352} a^{7} - \frac{6685}{17408} a^{6} + \frac{1087}{8704} a^{5} + \frac{8469}{69632} a^{4} + \frac{567}{8704} a^{3} + \frac{207}{512} a^{2} - \frac{45}{128} a + \frac{85}{256}$, $\frac{1}{557056} a^{19} - \frac{1}{139264} a^{18} + \frac{65}{139264} a^{17} - \frac{261}{34816} a^{16} + \frac{16887}{557056} a^{15} + \frac{11427}{69632} a^{14} - \frac{965}{557056} a^{13} + \frac{6117}{278528} a^{12} - \frac{218001}{557056} a^{11} + \frac{55771}{139264} a^{10} - \frac{231}{512} a^{9} + \frac{1273}{4096} a^{8} + \frac{69355}{139264} a^{7} - \frac{45}{17408} a^{6} - \frac{152795}{557056} a^{5} - \frac{105705}{278528} a^{4} - \frac{33191}{69632} a^{3} + \frac{815}{2048} a^{2} + \frac{569}{2048} a + \frac{363}{1024}$, $\frac{1}{4456448} a^{20} + \frac{1}{2228224} a^{19} - \frac{5}{1114112} a^{18} - \frac{327}{557056} a^{17} + \frac{12311}{4456448} a^{16} + \frac{2657}{131072} a^{15} - \frac{7221}{4456448} a^{14} - \frac{206517}{1114112} a^{13} + \frac{1002539}{4456448} a^{12} - \frac{545277}{2228224} a^{11} + \frac{65401}{557056} a^{10} - \frac{228295}{557056} a^{9} - \frac{549801}{1114112} a^{8} + \frac{29965}{557056} a^{7} - \frac{766619}{4456448} a^{6} - \frac{412093}{1114112} a^{5} + \frac{293047}{1114112} a^{4} + \frac{45701}{139264} a^{3} + \frac{2707}{16384} a^{2} - \frac{1109}{4096} a + \frac{321}{4096}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $16$ | 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: | \( 166388749255000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1959552 |
| The 333 conjugacy class representatives for t21n123 are not computed |
| Character table for t21n123 is not computed |
Intermediate fields
| 7.7.594823321.1 |
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 | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | $21$ | R | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ | $21$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ | $21$ | ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ | ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{9}$ |
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 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.24 | $x^{14} - 3 x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{2} + 2 x + 3$ | $2$ | $7$ | $14$ | 14T9 | $[2, 2, 2, 2]^{7}$ | |
| 3 | Data not computed | ||||||
| $17$ | $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| $29$ | 29.7.6.2 | $x^{7} - 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.14.12.1 | $x^{14} + 2407 x^{7} + 1839267$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ | |