Normalized defining polynomial
\( x^{21} - 66 x^{19} - 44 x^{18} + 1755 x^{17} + 2340 x^{16} - 23331 x^{15} - 48222 x^{14} + 149211 x^{13} + 476480 x^{12} - 237114 x^{11} - 2187516 x^{10} - 1872752 x^{9} + 3043296 x^{8} + 7287477 x^{7} + 4627874 x^{6} - 2062908 x^{5} - 5531544 x^{4} - 4256016 x^{3} - 1747872 x^{2} - 388416 x - 36992 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[19, 1]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-427377791081311923649394333215481320454012928=-\,2^{14}\cdot 3^{21}\cdot 17^{2}\cdot 29^{18}\cdot 41\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $133.44$ | 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: | $2, 3, 17, 29, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{3}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{16} + \frac{1}{32} a^{15} + \frac{13}{32} a^{14} + \frac{1}{8} a^{13} - \frac{21}{64} a^{12} + \frac{13}{32} a^{11} - \frac{7}{64} a^{10} - \frac{3}{16} a^{9} + \frac{11}{64} a^{8} + \frac{3}{32} a^{7} + \frac{5}{32} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{21}{64} a^{2} + \frac{3}{16} a + \frac{5}{16}$, $\frac{1}{8704} a^{17} - \frac{245}{4352} a^{15} - \frac{603}{2176} a^{14} + \frac{1435}{8704} a^{13} + \frac{161}{2176} a^{12} - \frac{2747}{8704} a^{11} - \frac{1919}{4352} a^{10} + \frac{2467}{8704} a^{9} + \frac{231}{544} a^{8} + \frac{927}{4352} a^{7} - \frac{759}{2176} a^{6} + \frac{151}{544} a^{5} - \frac{49}{136} a^{4} + \frac{3029}{8704} a^{3} - \frac{1999}{4352} a^{2} + \frac{15}{128} a + \frac{11}{64}$, $\frac{1}{69632} a^{18} - \frac{1}{34816} a^{17} - \frac{245}{34816} a^{16} - \frac{179}{8704} a^{15} + \frac{32371}{69632} a^{14} - \frac{14169}{34816} a^{13} - \frac{21443}{69632} a^{12} - \frac{881}{8704} a^{11} + \frac{18847}{69632} a^{10} - \frac{4971}{34816} a^{9} + \frac{14639}{34816} a^{8} + \frac{3509}{8704} a^{7} + \frac{2149}{8704} a^{6} + \frac{295}{2176} a^{5} + \frac{18005}{69632} a^{4} - \frac{169}{8704} a^{3} + \frac{2215}{8704} a^{2} + \frac{47}{128} a - \frac{75}{256}$, $\frac{1}{557056} a^{19} - \frac{1}{139264} a^{18} + \frac{13}{278528} a^{17} - \frac{113}{139264} a^{16} - \frac{397}{32768} a^{15} - \frac{627}{69632} a^{14} - \frac{65631}{557056} a^{13} - \frac{26113}{278528} a^{12} + \frac{19119}{557056} a^{11} + \frac{53883}{139264} a^{10} + \frac{64261}{278528} a^{9} + \frac{30267}{139264} a^{8} - \frac{8069}{69632} a^{7} - \frac{2967}{34816} a^{6} - \frac{156523}{557056} a^{5} - \frac{20729}{278528} a^{4} - \frac{3783}{69632} a^{3} - \frac{6697}{34816} a^{2} + \frac{377}{2048} a - \frac{309}{1024}$, $\frac{1}{4456448} a^{20} + \frac{1}{2228224} a^{19} + \frac{1}{2228224} a^{18} + \frac{27}{557056} a^{17} - \frac{9461}{4456448} a^{16} + \frac{130333}{2228224} a^{15} + \frac{2011665}{4456448} a^{14} + \frac{489969}{1114112} a^{13} - \frac{1078621}{4456448} a^{12} + \frac{18947}{2228224} a^{11} + \frac{563913}{2228224} a^{10} - \frac{9115}{557056} a^{9} - \frac{7221}{139264} a^{8} - \frac{18739}{139264} a^{7} + \frac{1346549}{4456448} a^{6} - \frac{492957}{1114112} a^{5} - \frac{192633}{1114112} a^{4} - \frac{65007}{139264} a^{3} + \frac{47187}{278528} a^{2} + \frac{379}{4096} a + \frac{801}{4096}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 2603642192310000 \) (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 | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ | $21$ | R | ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.17 | $x^{14} + x^{12} + 2 x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 1$ | $2$ | $7$ | $14$ | $C_2 \wr C_7$ | $[2, 2, 2, 2, 2, 2]^{14}$ | |
| 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 | $[\ ]$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| $29$ | 29.7.6.2 | $x^{7} - 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.7.6.2 | $x^{7} - 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 29.7.6.2 | $x^{7} - 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 41 | Data not computed | ||||||