Normalized defining polynomial
\( x^{21} - 84 x^{19} - 56 x^{18} + 2907 x^{17} + 3876 x^{16} - 52357 x^{15} - 107298 x^{14} + 498141 x^{13} + 1503232 x^{12} - 1956987 x^{11} - 10911882 x^{10} - 4065803 x^{9} + 34784676 x^{8} + 56724123 x^{7} - 3587546 x^{6} - 106446852 x^{5} - 146475144 x^{4} - 102597616 x^{3} - 41434848 x^{2} - 9207744 x - 876928 \)
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: | \(3938561358012927168760351207095524681785277184000000=2^{14}\cdot 3^{21}\cdot 5^{6}\cdot 13^{2}\cdot 17^{2}\cdot 31^{2}\cdot 353\cdot 6679^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $286.37$ | 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: | $2, 3, 5, 13, 17, 31, 353, 6679$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} + \frac{3}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{1}{8} a^{14} + \frac{3}{8} a^{13} + \frac{27}{64} a^{12} + \frac{13}{32} a^{11} - \frac{9}{64} a^{10} - \frac{5}{16} a^{9} + \frac{13}{64} a^{8} - \frac{11}{32} a^{7} - \frac{31}{64} a^{6} + \frac{3}{8} a^{5} + \frac{13}{64} a^{4} + \frac{7}{32} a^{3} + \frac{31}{64} a^{2} - \frac{7}{16} a - \frac{1}{16}$, $\frac{1}{512} a^{17} + \frac{1}{128} a^{15} + \frac{17}{64} a^{14} - \frac{21}{512} a^{13} + \frac{57}{128} a^{12} - \frac{253}{512} a^{11} + \frac{127}{256} a^{10} + \frac{245}{512} a^{9} + \frac{5}{32} a^{8} - \frac{51}{512} a^{7} + \frac{43}{256} a^{6} - \frac{163}{512} a^{5} - \frac{3}{128} a^{4} + \frac{131}{512} a^{3} - \frac{13}{256} a^{2} + \frac{29}{128} a + \frac{17}{64}$, $\frac{1}{4096} a^{18} - \frac{1}{2048} a^{17} + \frac{1}{1024} a^{16} + \frac{1}{32} a^{15} - \frac{1829}{4096} a^{14} - \frac{377}{2048} a^{13} + \frac{1339}{4096} a^{12} - \frac{33}{512} a^{11} - \frac{263}{4096} a^{10} + \frac{819}{2048} a^{9} - \frac{723}{4096} a^{8} - \frac{209}{1024} a^{7} - \frac{335}{4096} a^{6} - \frac{355}{2048} a^{5} - \frac{1381}{4096} a^{4} - \frac{25}{128} a^{3} - \frac{171}{512} a^{2} + \frac{45}{128} a - \frac{81}{256}$, $\frac{1}{32768} a^{19} - \frac{1}{8192} a^{18} + \frac{1}{4096} a^{17} + \frac{15}{4096} a^{16} + \frac{2011}{32768} a^{15} + \frac{875}{4096} a^{14} + \frac{6943}{32768} a^{13} - \frac{7615}{16384} a^{12} + \frac{12553}{32768} a^{11} - \frac{483}{8192} a^{10} - \frac{16287}{32768} a^{9} + \frac{6449}{16384} a^{8} + \frac{5433}{32768} a^{7} + \frac{2043}{4096} a^{6} - \frac{12249}{32768} a^{5} - \frac{5163}{16384} a^{4} + \frac{1053}{4096} a^{3} + \frac{773}{2048} a^{2} + \frac{1019}{2048} a - \frac{175}{1024}$, $\frac{1}{262144} a^{20} + \frac{1}{131072} a^{19} - \frac{1}{16384} a^{18} + \frac{21}{32768} a^{17} - \frac{1365}{262144} a^{16} + \frac{5437}{131072} a^{15} - \frac{114897}{262144} a^{14} - \frac{17969}{65536} a^{13} - \frac{25579}{262144} a^{12} - \frac{16555}{131072} a^{11} - \frac{56551}{262144} a^{10} - \frac{363}{32768} a^{9} - \frac{35963}{262144} a^{8} - \frac{12393}{131072} a^{7} - \frac{82121}{262144} a^{6} - \frac{20955}{65536} a^{5} + \frac{30649}{65536} a^{4} + \frac{599}{4096} a^{3} + \frac{5913}{16384} a^{2} + \frac{1697}{4096} a - \frac{1293}{4096}$
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: | \( 17342526476000000000 \) (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
| 7.7.1115226025.1 |
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 | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | R | R | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ | $21$ | ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.1 | $x^{14} + 3 x^{12} - 2 x^{11} - 2 x^{10} + 4 x^{9} + 2 x^{7} + 2 x^{5} + 2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x - 3$ | $2$ | $7$ | $14$ | $C_2 \wr C_7$ | $[2, 2, 2, 2, 2, 2, 2]^{7}$ | |
| 3 | Data not computed | ||||||
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.9.0.1 | $x^{9} + x^{2} - 2 x + 2$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| $13$ | 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.8.0.1 | $x^{8} + 4 x^{2} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $17$ | 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 17.6.0.1 | $x^{6} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 17.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 31 | Data not computed | ||||||
| 353 | Data not computed | ||||||
| 6679 | Data not computed | ||||||