Normalized defining polynomial
\( x^{21} - 2 x^{20} - 34 x^{19} + 44 x^{18} + 440 x^{17} - 336 x^{16} - 2735 x^{15} + 1258 x^{14} + 9114 x^{13} - 2548 x^{12} - 17108 x^{11} + 2708 x^{10} + 18395 x^{9} - 1186 x^{8} - 11144 x^{7} - 160 x^{6} + 3584 x^{5} + 284 x^{4} - 539 x^{3} - 66 x^{2} + 28 x + 4 \)
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: | \(114304429232803182734215052479732645888=2^{32}\cdot 37^{7}\cdot 809^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $64.91$ | 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: | $2, 37, 809$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{254581539173186637878} a^{20} + \frac{7735471698135525563}{127290769586593318939} a^{19} + \frac{19345533267848492272}{127290769586593318939} a^{18} - \frac{42249580609372592937}{127290769586593318939} a^{17} + \frac{32248825226002649802}{127290769586593318939} a^{16} - \frac{13597119786106395543}{127290769586593318939} a^{15} - \frac{58327804629339979811}{254581539173186637878} a^{14} + \frac{2101033014069154594}{127290769586593318939} a^{13} - \frac{40710486289356312323}{127290769586593318939} a^{12} - \frac{12662386509079076768}{127290769586593318939} a^{11} + \frac{45121579397493201787}{127290769586593318939} a^{10} + \frac{14578446841158565069}{127290769586593318939} a^{9} - \frac{121794114498656913227}{254581539173186637878} a^{8} + \frac{61692511844016917248}{127290769586593318939} a^{7} + \frac{5400995876832554520}{127290769586593318939} a^{6} + \frac{3875107887303004766}{127290769586593318939} a^{5} - \frac{30894598403576919432}{127290769586593318939} a^{4} + \frac{19392480810457944675}{127290769586593318939} a^{3} + \frac{49023595495642578429}{254581539173186637878} a^{2} - \frac{34477219498767380154}{127290769586593318939} a - \frac{40088340622463099590}{127290769586593318939}$
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: | \( 8566215956620 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1008 |
| The 18 conjugacy class representatives for t21n27 |
| Character table for t21n27 |
Intermediate fields
| 3.3.148.1, 7.7.670188544.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 24 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently 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 | $21$ | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | $21$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ | ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | R | $21$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.10.2 | $x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{2} + 2$ | $6$ | $1$ | $10$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 2.12.20.37 | $x^{12} - 6 x^{10} - x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} - 7$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 37 | Data not computed | ||||||
| 809 | Data not computed | ||||||