Normalized defining polynomial
\( x^{22} - 28 x^{20} + 325 x^{18} - 2049 x^{16} + 7734 x^{14} - 18224 x^{12} + 27221 x^{10} - 25625 x^{8} + 14751 x^{6} - 4857 x^{4} + 797 x^{2} - 47 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(338309725595529099314140070845349888=2^{22}\cdot 23^{20}\cdot 47\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $41.21$ | 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: | $2, 23, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} - \frac{3}{47} a^{16} - \frac{4}{47} a^{14} + \frac{23}{47} a^{12} + \frac{19}{47} a^{10} + \frac{3}{47} a^{8} + \frac{4}{47} a^{6} - \frac{14}{47} a^{4} - \frac{10}{47} a^{2}$, $\frac{1}{47} a^{19} - \frac{3}{47} a^{17} - \frac{4}{47} a^{15} + \frac{23}{47} a^{13} + \frac{19}{47} a^{11} + \frac{3}{47} a^{9} + \frac{4}{47} a^{7} - \frac{14}{47} a^{5} - \frac{10}{47} a^{3}$, $\frac{1}{895021} a^{20} - \frac{15}{6439} a^{18} + \frac{23538}{895021} a^{16} + \frac{178021}{895021} a^{14} - \frac{60745}{895021} a^{12} - \frac{25925}{895021} a^{10} + \frac{53589}{895021} a^{8} - \frac{437217}{895021} a^{6} - \frac{183114}{895021} a^{4} - \frac{390759}{895021} a^{2} - \frac{5138}{19043}$, $\frac{1}{895021} a^{21} - \frac{15}{6439} a^{19} + \frac{23538}{895021} a^{17} + \frac{178021}{895021} a^{15} - \frac{60745}{895021} a^{13} - \frac{25925}{895021} a^{11} + \frac{53589}{895021} a^{9} - \frac{437217}{895021} a^{7} - \frac{183114}{895021} a^{5} - \frac{390759}{895021} a^{3} - \frac{5138}{19043} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $21$ | 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: | \( 45337860614.1 \) (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 | R | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | R | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |
| $47$ | $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | 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 | $[\ ]$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |