Normalized defining polynomial
\( x^{21} - 27 x^{19} - 18 x^{18} + 234 x^{17} + 312 x^{16} - 409 x^{15} - 1026 x^{14} - 2547 x^{13} - 5120 x^{12} + 621 x^{11} + 16422 x^{10} + 28117 x^{9} + 31140 x^{8} + 25446 x^{7} + 1924 x^{6} - 30024 x^{5} - 42480 x^{4} - 29920 x^{3} - 12096 x^{2} - 2688 x - 256 \)
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: | \(-330985239322495312738299332696447755321344=-\,2^{15}\cdot 3^{21}\cdot 149^{6}\cdot 211^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $94.87$ | 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: | $2, 3, 149, 211$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{15} + \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{1}{64} a^{14} - \frac{1}{4} a^{13} - \frac{7}{32} a^{12} - \frac{5}{16} a^{11} + \frac{31}{64} a^{10} + \frac{7}{16} a^{9} + \frac{13}{64} a^{8} - \frac{11}{32} a^{7} + \frac{9}{64} a^{6} + \frac{3}{8} a^{5} - \frac{19}{64} a^{4} + \frac{7}{32} a^{3} - \frac{11}{32} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{512} a^{17} - \frac{3}{512} a^{15} - \frac{41}{256} a^{14} + \frac{105}{256} a^{13} + \frac{9}{64} a^{12} + \frac{7}{512} a^{11} + \frac{111}{256} a^{10} + \frac{21}{512} a^{9} + \frac{9}{32} a^{8} + \frac{53}{512} a^{7} + \frac{3}{256} a^{6} + \frac{189}{512} a^{5} - \frac{3}{128} a^{4} - \frac{57}{256} a^{3} + \frac{17}{128} a^{2} + \frac{15}{64} a + \frac{11}{32}$, $\frac{1}{4096} a^{18} - \frac{1}{2048} a^{17} - \frac{3}{4096} a^{16} - \frac{19}{1024} a^{15} + \frac{187}{2048} a^{14} - \frac{471}{1024} a^{13} + \frac{375}{4096} a^{12} + \frac{45}{256} a^{11} - \frac{423}{4096} a^{10} + \frac{563}{2048} a^{9} + \frac{1301}{4096} a^{8} + \frac{359}{1024} a^{7} - \frac{335}{4096} a^{6} + \frac{317}{2048} a^{5} - \frac{301}{2048} a^{4} - \frac{91}{512} a^{3} + \frac{95}{256} a^{2} - \frac{9}{64} a - \frac{43}{128}$, $\frac{1}{32768} a^{19} - \frac{1}{8192} a^{18} + \frac{1}{32768} a^{17} - \frac{35}{16384} a^{16} + \frac{263}{16384} a^{15} - \frac{329}{4096} a^{14} - \frac{16337}{32768} a^{13} - \frac{15}{16384} a^{12} - \frac{1863}{32768} a^{11} + \frac{493}{8192} a^{10} + \frac{3145}{32768} a^{9} - \frac{2631}{16384} a^{8} - \frac{11399}{32768} a^{7} + \frac{1699}{4096} a^{6} - \frac{935}{16384} a^{5} + \frac{2167}{8192} a^{4} + \frac{349}{1024} a^{3} - \frac{497}{1024} a^{2} + \frac{121}{1024} a + \frac{171}{512}$, $\frac{1}{262144} a^{20} + \frac{1}{131072} a^{19} - \frac{23}{262144} a^{18} - \frac{1}{4096} a^{17} + \frac{53}{131072} a^{16} + \frac{131}{65536} a^{15} + \frac{639}{262144} a^{14} + \frac{63}{65536} a^{13} - \frac{2043}{262144} a^{12} - \frac{4603}{131072} a^{11} - \frac{17791}{262144} a^{10} - \frac{2395}{32768} a^{9} - \frac{10203}{262144} a^{8} + \frac{5367}{131072} a^{7} + \frac{23457}{131072} a^{6} + \frac{11969}{32768} a^{5} - \frac{12583}{32768} a^{4} + \frac{573}{8192} a^{3} + \frac{211}{8192} a^{2} + \frac{11}{2048} a + \frac{1}{2048}$
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: | \( 371198961249000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 705438720 |
| The 246 conjugacy class representatives for t21n151 are not computed |
| Character table for t21n151 is not computed |
Intermediate fields
| 7.7.988410721.1 |
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 | $21$ | $21$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.10.8 | $x^{10} + x^{8} - 2 x^{6} - 2 x^{4} + x^{2} + 33$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
| 3 | Data not computed | ||||||
| 149 | Data not computed | ||||||
| 211 | Data not computed | ||||||