Normalized defining polynomial
\( x^{21} - 63 x^{19} - 42 x^{18} + 1665 x^{17} + 2220 x^{16} - 23155 x^{15} - 47790 x^{14} + 168858 x^{13} + 528168 x^{12} - 449145 x^{11} - 3043422 x^{10} - 1732173 x^{9} + 7496316 x^{8} + 13524411 x^{7} + 1957846 x^{6} - 18965124 x^{5} - 27585864 x^{4} - 19533296 x^{3} - 7904736 x^{2} - 1756608 x - 167296 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-112907939754257152540470809754816990560256=-\,2^{14}\cdot 3^{21}\cdot 23^{3}\cdot 29\cdot 239^{3}\cdot 431^{3}\cdot 1307^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $90.13$ | 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: | $2, 3, 23, 29, 239, 431, 1307$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} - \frac{3}{8} a^{13} + \frac{1}{4} a^{12} - \frac{3}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{29}{64} a^{14} + \frac{1}{4} a^{13} - \frac{23}{64} a^{12} + \frac{7}{32} a^{11} - \frac{15}{64} a^{10} - \frac{3}{16} a^{9} - \frac{11}{32} a^{8} - \frac{5}{16} a^{7} - \frac{1}{64} a^{6} - \frac{5}{64} a^{4} - \frac{15}{32} a^{3} + \frac{23}{64} a^{2} + \frac{1}{16} a + \frac{7}{16}$, $\frac{1}{512} a^{17} + \frac{25}{512} a^{15} + \frac{75}{256} a^{14} + \frac{73}{512} a^{13} + \frac{63}{128} a^{12} + \frac{213}{512} a^{11} + \frac{105}{256} a^{10} + \frac{65}{256} a^{9} + \frac{11}{64} a^{8} + \frac{167}{512} a^{7} + \frac{97}{256} a^{6} - \frac{133}{512} a^{5} + \frac{59}{128} a^{4} - \frac{45}{512} a^{3} - \frac{53}{256} a^{2} - \frac{59}{128} a + \frac{25}{64}$, $\frac{1}{4096} a^{18} - \frac{1}{2048} a^{17} + \frac{25}{4096} a^{16} + \frac{25}{1024} a^{15} - \frac{1251}{4096} a^{14} + \frac{821}{2048} a^{13} - \frac{1315}{4096} a^{12} - \frac{155}{512} a^{11} + \frac{367}{2048} a^{10} - \frac{427}{1024} a^{9} + \frac{1015}{4096} a^{8} - \frac{35}{1024} a^{7} - \frac{521}{4096} a^{6} - \frac{517}{2048} a^{5} + \frac{1531}{4096} a^{4} + \frac{63}{256} a^{3} - \frac{67}{512} a^{2} - \frac{43}{128} a + \frac{103}{256}$, $\frac{1}{32768} a^{19} - \frac{1}{8192} a^{18} + \frac{29}{32768} a^{17} + \frac{25}{16384} a^{16} - \frac{1451}{32768} a^{15} - \frac{509}{2048} a^{14} + \frac{11785}{32768} a^{13} - \frac{1353}{16384} a^{12} + \frac{5703}{16384} a^{11} + \frac{1651}{4096} a^{10} + \frac{4431}{32768} a^{9} + \frac{7107}{16384} a^{8} - \frac{8433}{32768} a^{7} - \frac{1535}{4096} a^{6} - \frac{12785}{32768} a^{5} + \frac{7165}{16384} a^{4} + \frac{1217}{4096} a^{3} - \frac{787}{2048} a^{2} + \frac{787}{2048} a - \frac{103}{1024}$, $\frac{1}{262144} a^{20} + \frac{1}{131072} a^{19} + \frac{5}{262144} a^{18} + \frac{7}{8192} a^{17} - \frac{1151}{262144} a^{16} + \frac{7959}{131072} a^{15} + \frac{126761}{262144} a^{14} + \frac{617}{65536} a^{13} - \frac{51567}{131072} a^{12} + \frac{4027}{65536} a^{11} + \frac{116447}{262144} a^{10} - \frac{2821}{8192} a^{9} + \frac{109619}{262144} a^{8} - \frac{64207}{131072} a^{7} - \frac{86465}{262144} a^{6} - \frac{15595}{65536} a^{5} + \frac{7545}{65536} a^{4} - \frac{205}{1024} a^{3} + \frac{2209}{16384} a^{2} - \frac{1431}{4096} a - \frac{1333}{4096}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 16402886226500 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1410877440 |
| The 429 conjugacy class representatives for t21n152 are not computed |
| Character table for t21n152 is not computed |
Intermediate fields
| 7.5.2369207.1 |
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 | ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ | $21$ | $21$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | R | R | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.18 | $x^{14} + 4 x^{13} + 3 x^{12} + 2 x^{11} - 2 x^{9} + 2 x^{7} - 2 x^{4} - 2 x^{3} - 2 x^{2} + 1$ | $2$ | $7$ | $14$ | $C_2 \wr C_7$ | $[2, 2, 2, 2, 2, 2, 2]^{7}$ | |
| 3 | Data not computed | ||||||
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 23.6.0.1 | $x^{6} - x + 15$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 29 | Data not computed | ||||||
| 239 | Data not computed | ||||||
| 431 | Data not computed | ||||||
| 1307 | Data not computed | ||||||