Normalized defining polynomial
\( x^{22} - x^{21} + 23 x^{20} - 21 x^{19} + 261 x^{18} - 217 x^{17} + 1915 x^{16} - 1441 x^{15} + 10049 x^{14} - 6777 x^{13} + 39495 x^{12} - 23525 x^{11} + 118485 x^{10} - 60993 x^{9} + 271323 x^{8} - 116721 x^{7} + 465345 x^{6} - 158193 x^{5} + 570807 x^{4} - 137781 x^{3} + 452709 x^{2} - 59049 x + 177147 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-170726762755621223297063838384459279239=-\,23^{20}\cdot 47\cdot 2116640137\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $54.68$ | 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: | $23, 47, 2116640137$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{4}{9} a^{11} - \frac{1}{3} a^{10} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{4}{27} a^{12} + \frac{2}{9} a^{11} - \frac{1}{3} a^{10} - \frac{1}{27} a^{9} - \frac{2}{27} a^{8} - \frac{10}{27} a^{7} + \frac{5}{27} a^{6} - \frac{2}{9} a^{4} - \frac{8}{27} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{81} a^{15} - \frac{1}{81} a^{14} - \frac{4}{81} a^{13} + \frac{2}{27} a^{12} - \frac{4}{9} a^{11} + \frac{26}{81} a^{10} - \frac{29}{81} a^{9} - \frac{37}{81} a^{8} - \frac{22}{81} a^{7} - \frac{1}{3} a^{6} - \frac{2}{27} a^{5} - \frac{35}{81} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{243} a^{16} - \frac{1}{243} a^{15} - \frac{4}{243} a^{14} + \frac{2}{81} a^{13} - \frac{4}{27} a^{12} + \frac{26}{243} a^{11} - \frac{110}{243} a^{10} + \frac{44}{243} a^{9} - \frac{103}{243} a^{8} - \frac{1}{9} a^{7} + \frac{25}{81} a^{6} - \frac{35}{243} a^{5} - \frac{2}{27} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{729} a^{17} - \frac{1}{729} a^{16} - \frac{4}{729} a^{15} + \frac{2}{243} a^{14} - \frac{4}{81} a^{13} + \frac{26}{729} a^{12} + \frac{133}{729} a^{11} - \frac{199}{729} a^{10} - \frac{103}{729} a^{9} - \frac{10}{27} a^{8} + \frac{25}{243} a^{7} + \frac{208}{729} a^{6} - \frac{2}{81} a^{5} - \frac{10}{27} a^{4} + \frac{5}{27} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{2187} a^{18} - \frac{1}{2187} a^{17} - \frac{4}{2187} a^{16} + \frac{2}{729} a^{15} - \frac{4}{243} a^{14} + \frac{26}{2187} a^{13} + \frac{133}{2187} a^{12} - \frac{199}{2187} a^{11} - \frac{832}{2187} a^{10} - \frac{37}{81} a^{9} - \frac{218}{729} a^{8} + \frac{937}{2187} a^{7} + \frac{79}{243} a^{6} - \frac{37}{81} a^{5} + \frac{32}{81} a^{4} + \frac{4}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{6561} a^{19} - \frac{1}{6561} a^{18} - \frac{4}{6561} a^{17} + \frac{2}{2187} a^{16} - \frac{4}{729} a^{15} + \frac{26}{6561} a^{14} + \frac{133}{6561} a^{13} - \frac{199}{6561} a^{12} - \frac{832}{6561} a^{11} + \frac{44}{243} a^{10} + \frac{511}{2187} a^{9} + \frac{3124}{6561} a^{8} + \frac{322}{729} a^{7} + \frac{44}{243} a^{6} + \frac{113}{243} a^{5} + \frac{13}{27} a^{4} + \frac{13}{27} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{19683} a^{20} - \frac{1}{19683} a^{19} - \frac{4}{19683} a^{18} + \frac{2}{6561} a^{17} - \frac{4}{2187} a^{16} + \frac{26}{19683} a^{15} + \frac{133}{19683} a^{14} - \frac{199}{19683} a^{13} - \frac{832}{19683} a^{12} + \frac{44}{729} a^{11} + \frac{2698}{6561} a^{10} + \frac{9685}{19683} a^{9} + \frac{1051}{2187} a^{8} + \frac{287}{729} a^{7} + \frac{113}{729} a^{6} + \frac{13}{81} a^{5} + \frac{13}{81} a^{4} - \frac{1}{27} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{59049} a^{21} - \frac{1}{59049} a^{20} - \frac{4}{59049} a^{19} + \frac{2}{19683} a^{18} - \frac{4}{6561} a^{17} + \frac{26}{59049} a^{16} + \frac{133}{59049} a^{15} - \frac{199}{59049} a^{14} - \frac{832}{59049} a^{13} + \frac{44}{2187} a^{12} + \frac{9259}{19683} a^{11} + \frac{29368}{59049} a^{10} + \frac{1051}{6561} a^{9} - \frac{442}{2187} a^{8} - \frac{616}{2187} a^{7} - \frac{68}{243} a^{6} - \frac{68}{243} a^{5} + \frac{26}{81} a^{4} - \frac{10}{27} a^{3} - \frac{1}{3} a^{2}$
Class group and class number
$C_{6878}$, which has order $6878$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 1038656.82438 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 22528 |
| The 208 conjugacy class representatives for t22n28 are not computed |
| Character table for t22n28 is not computed |
Intermediate fields
| \(\Q(\zeta_{23})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | $22$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 23 | Data not computed | ||||||
| $47$ | $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2116640137 | Data not computed | ||||||