Normalized defining polynomial
\( x^{21} - 48 x^{19} - 32 x^{18} + 909 x^{17} + 1212 x^{16} - 8236 x^{15} - 17280 x^{14} + 31653 x^{13} + 112568 x^{12} + 8208 x^{11} - 305232 x^{10} - 359509 x^{9} + 111852 x^{8} + 580389 x^{7} + 488282 x^{6} + 52164 x^{5} - 210168 x^{4} - 187152 x^{3} - 78624 x^{2} - 17472 x - 1664 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[17, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1437253422311392754840506463696646924091392=2^{14}\cdot 3^{21}\cdot 7\cdot 13^{2}\cdot 577^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $101.74$ | 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: | $2, 3, 7, 13, 577$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{3}{8} a^{11} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} - \frac{5}{16} a^{14} - \frac{1}{8} a^{13} - \frac{19}{64} a^{12} + \frac{11}{32} a^{11} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{21}{64} a^{8} - \frac{15}{32} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{21}{64} a^{4} + \frac{1}{32} a^{3} - \frac{15}{64} a^{2} + \frac{7}{16} a + \frac{1}{16}$, $\frac{1}{512} a^{17} - \frac{3}{64} a^{15} - \frac{3}{16} a^{14} - \frac{131}{512} a^{13} - \frac{17}{128} a^{12} - \frac{61}{128} a^{11} + \frac{7}{16} a^{10} - \frac{203}{512} a^{9} + \frac{31}{64} a^{8} - \frac{23}{64} a^{7} - \frac{5}{32} a^{6} + \frac{91}{512} a^{5} - \frac{5}{128} a^{4} - \frac{211}{512} a^{3} - \frac{35}{256} a^{2} + \frac{3}{128} a - \frac{17}{64}$, $\frac{1}{4096} a^{18} - \frac{1}{2048} a^{17} - \frac{3}{512} a^{16} - \frac{3}{256} a^{15} + \frac{573}{4096} a^{14} - \frac{159}{2048} a^{13} + \frac{229}{1024} a^{12} + \frac{217}{512} a^{11} + \frac{885}{4096} a^{10} + \frac{327}{2048} a^{9} - \frac{85}{512} a^{8} - \frac{23}{128} a^{7} + \frac{1787}{4096} a^{6} - \frac{357}{2048} a^{5} + \frac{341}{4096} a^{4} + \frac{11}{128} a^{3} + \frac{211}{512} a^{2} - \frac{53}{128} a + \frac{81}{256}$, $\frac{1}{32768} a^{19} - \frac{1}{8192} a^{18} - \frac{5}{8192} a^{17} + \frac{669}{32768} a^{15} - \frac{183}{4096} a^{14} + \frac{353}{2048} a^{13} + \frac{253}{1024} a^{12} + \frac{1509}{32768} a^{11} + \frac{1769}{8192} a^{10} - \frac{497}{8192} a^{9} - \frac{985}{2048} a^{8} + \frac{7355}{32768} a^{7} - \frac{131}{512} a^{6} + \frac{14057}{32768} a^{5} + \frac{1883}{16384} a^{4} + \frac{1147}{4096} a^{3} - \frac{317}{2048} a^{2} + \frac{549}{2048} a - \frac{337}{1024}$, $\frac{1}{262144} a^{20} + \frac{1}{131072} a^{19} + \frac{5}{65536} a^{18} - \frac{31}{32768} a^{17} - \frac{867}{262144} a^{16} - \frac{261}{131072} a^{15} + \frac{131}{1024} a^{14} - \frac{343}{2048} a^{13} + \frac{43173}{262144} a^{12} - \frac{34687}{131072} a^{11} - \frac{8491}{65536} a^{10} + \frac{9963}{32768} a^{9} - \frac{65189}{262144} a^{8} + \frac{10705}{131072} a^{7} + \frac{45353}{262144} a^{6} + \frac{10603}{65536} a^{5} - \frac{2985}{65536} a^{4} + \frac{109}{4096} a^{3} - \frac{6889}{16384} a^{2} - \frac{1553}{4096} a + \frac{1309}{4096}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $18$ | 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: | \( 101941065977000 \) (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
| 7.7.192100033.1 |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ | R | $21$ | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ | $21$ |
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 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.29 | $x^{14} + 2 x^{13} - x^{12} + 2 x^{7} + 2 x^{5} + 2 x^{3} + 2 x - 1$ | $2$ | $7$ | $14$ | $C_2 \wr C_7$ | $[2, 2, 2, 2, 2, 2]^{14}$ | |
| 3 | Data not computed | ||||||
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 577 | Data not computed | ||||||