Normalized defining polynomial
\( x^{22} - 9 x^{21} + 29 x^{20} - 38 x^{19} - 4 x^{18} + 86 x^{17} - 143 x^{16} + 138 x^{15} - 79 x^{14} - 37 x^{13} + 184 x^{12} - 257 x^{11} + 184 x^{10} - 37 x^{9} - 79 x^{8} + 138 x^{7} - 143 x^{6} + 86 x^{5} - 4 x^{4} - 38 x^{3} + 29 x^{2} - 9 x + 1 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-80659324072725558117423074447=-\,23^{20}\cdot 47\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.60$ | 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: | $23, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} + \frac{5}{47} a^{17} + \frac{13}{47} a^{16} + \frac{10}{47} a^{15} - \frac{20}{47} a^{14} + \frac{16}{47} a^{13} + \frac{1}{47} a^{12} - \frac{10}{47} a^{11} - \frac{14}{47} a^{10} + \frac{14}{47} a^{9} - \frac{14}{47} a^{8} - \frac{10}{47} a^{7} + \frac{1}{47} a^{6} + \frac{16}{47} a^{5} - \frac{20}{47} a^{4} + \frac{10}{47} a^{3} + \frac{13}{47} a^{2} + \frac{5}{47} a + \frac{1}{47}$, $\frac{1}{47} a^{19} - \frac{12}{47} a^{17} - \frac{8}{47} a^{16} - \frac{23}{47} a^{15} + \frac{22}{47} a^{14} + \frac{15}{47} a^{13} - \frac{15}{47} a^{12} - \frac{11}{47} a^{11} - \frac{10}{47} a^{10} + \frac{10}{47} a^{9} + \frac{13}{47} a^{8} + \frac{4}{47} a^{7} + \frac{11}{47} a^{6} - \frac{6}{47} a^{5} + \frac{16}{47} a^{4} + \frac{10}{47} a^{3} - \frac{13}{47} a^{2} + \frac{23}{47} a - \frac{5}{47}$, $\frac{1}{47} a^{20} + \frac{5}{47} a^{17} - \frac{8}{47} a^{16} + \frac{1}{47} a^{15} + \frac{10}{47} a^{14} - \frac{11}{47} a^{13} + \frac{1}{47} a^{12} + \frac{11}{47} a^{11} - \frac{17}{47} a^{10} - \frac{7}{47} a^{9} - \frac{23}{47} a^{8} - \frac{15}{47} a^{7} + \frac{6}{47} a^{6} + \frac{20}{47} a^{5} + \frac{5}{47} a^{4} + \frac{13}{47} a^{3} - \frac{9}{47} a^{2} + \frac{8}{47} a + \frac{12}{47}$, $\frac{1}{47} a^{21} + \frac{14}{47} a^{17} - \frac{17}{47} a^{16} + \frac{7}{47} a^{15} - \frac{5}{47} a^{14} + \frac{15}{47} a^{13} + \frac{6}{47} a^{12} - \frac{14}{47} a^{11} + \frac{16}{47} a^{10} + \frac{1}{47} a^{9} + \frac{8}{47} a^{8} + \frac{9}{47} a^{7} + \frac{15}{47} a^{6} + \frac{19}{47} a^{5} + \frac{19}{47} a^{4} - \frac{12}{47} a^{3} - \frac{10}{47} a^{2} - \frac{13}{47} a - \frac{5}{47}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 918526.809002 \) (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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 | $[\ ]$ | |
| 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 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.1 | $x^{2} - 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |