↑

Properties

Label 22.16.1781764468...3423.1
Degree $22$
Signature $[16, 3]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, -24, 213, -74, -1217, 2226, -185, -3904, 5279, -1546, -3588, 4953, -2284, -804, 1782, -1077, 216, 128, -125, 51, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 51*x^20 - 125*x^19 + 128*x^18 + 216*x^17 - 1077*x^16 + 1782*x^15 - 804*x^14 - 2284*x^13 + 4953*x^12 - 3588*x^11 - 1546*x^10 + 5279*x^9 - 3904*x^8 - 185*x^7 + 2226*x^6 - 1217*x^5 - 74*x^4 + 213*x^3 - 24*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^22 - 11*x^21 + 51*x^20 - 125*x^19 + 128*x^18 + 216*x^17 - 1077*x^16 + 1782*x^15 - 804*x^14 - 2284*x^13 + 4953*x^12 - 3588*x^11 - 1546*x^10 + 5279*x^9 - 3904*x^8 - 185*x^7 + 2226*x^6 - 1217*x^5 - 74*x^4 + 213*x^3 - 24*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 51 x^{20} - 125 x^{19} + 128 x^{18} + 216 x^{17} - 1077 x^{16} + 1782 x^{15} - 804 x^{14} - 2284 x^{13} + 4953 x^{12} - 3588 x^{11} - 1546 x^{10} + 5279 x^{9} - 3904 x^{8} - 185 x^{7} + 2226 x^{6} - 1217 x^{5} - 74 x^{4} + 213 x^{3} - 24 x^{2} - 10 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{47} a^{16} - \frac{8}{47} a^{15} + \frac{11}{47} a^{14} + \frac{16}{47} a^{13} - \frac{15}{47} a^{12} - \frac{12}{47} a^{11} - \frac{10}{47} a^{10} + \frac{22}{47} a^{9} - \frac{22}{47} a^{8} - \frac{1}{47} a^{7} - \frac{10}{47} a^{6} - \frac{4}{47} a^{5} + \frac{19}{47} a^{4} + \frac{23}{47} a^{3} - \frac{1}{47} a^{2} - \frac{9}{47} a - \frac{10}{47}$, $\frac{1}{47} a^{17} - \frac{6}{47} a^{15} + \frac{10}{47} a^{14} + \frac{19}{47} a^{13} + \frac{9}{47} a^{12} - \frac{12}{47} a^{11} - \frac{11}{47} a^{10} + \frac{13}{47} a^{9} + \frac{11}{47} a^{8} - \frac{18}{47} a^{7} + \frac{10}{47} a^{6} - \frac{13}{47} a^{5} - \frac{13}{47} a^{4} - \frac{5}{47} a^{3} - \frac{17}{47} a^{2} + \frac{12}{47} a + \frac{14}{47}$, $\frac{1}{47} a^{18} + \frac{9}{47} a^{15} - \frac{9}{47} a^{14} + \frac{11}{47} a^{13} - \frac{8}{47} a^{12} + \frac{11}{47} a^{11} + \frac{2}{47} a^{9} - \frac{9}{47} a^{8} + \frac{4}{47} a^{7} + \frac{21}{47} a^{6} + \frac{10}{47} a^{5} + \frac{15}{47} a^{4} - \frac{20}{47} a^{3} + \frac{6}{47} a^{2} + \frac{7}{47} a - \frac{13}{47}$, $\frac{1}{47} a^{19} + \frac{16}{47} a^{15} + \frac{6}{47} a^{14} - \frac{11}{47} a^{13} + \frac{5}{47} a^{12} + \frac{14}{47} a^{11} - \frac{2}{47} a^{10} - \frac{19}{47} a^{9} + \frac{14}{47} a^{8} - \frac{17}{47} a^{7} + \frac{6}{47} a^{6} + \frac{4}{47} a^{5} - \frac{3}{47} a^{4} - \frac{13}{47} a^{3} + \frac{16}{47} a^{2} + \frac{21}{47} a - \frac{4}{47}$, $\frac{1}{307051} a^{20} - \frac{10}{307051} a^{19} + \frac{2003}{307051} a^{18} + \frac{1857}{307051} a^{17} - \frac{995}{307051} a^{16} - \frac{15825}{307051} a^{15} - \frac{146385}{307051} a^{14} + \frac{47553}{307051} a^{13} - \frac{126733}{307051} a^{12} + \frac{133516}{307051} a^{11} + \frac{49374}{307051} a^{10} + \frac{11279}{307051} a^{9} + \frac{29796}{307051} a^{8} + \frac{17603}{307051} a^{7} - \frac{16498}{307051} a^{6} - \frac{111030}{307051} a^{5} + \frac{36861}{307051} a^{4} - \frac{58}{2209} a^{3} - \frac{39829}{307051} a^{2} - \frac{60466}{307051} a + \frac{27687}{307051}$, $\frac{1}{307051} a^{21} + \frac{1903}{307051} a^{19} + \frac{2288}{307051} a^{18} - \frac{2024}{307051} a^{17} + \frac{357}{307051} a^{16} + \frac{41614}{307051} a^{15} + \frac{79760}{307051} a^{14} - \frac{128112}{307051} a^{13} - \frac{10138}{307051} a^{12} - \frac{137655}{307051} a^{11} + \frac{152237}{307051} a^{10} + \frac{116454}{307051} a^{9} + \frac{8512}{307051} a^{8} + \frac{100735}{307051} a^{7} + \frac{83305}{307051} a^{6} + \frac{109034}{307051} a^{5} - \frac{103295}{307051} a^{4} + \frac{49409}{307051} a^{3} + \frac{37752}{307051} a^{2} + \frac{43662}{307051} a - \frac{4049}{307051}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $18$
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:  \( 238425084.122 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

Degree 22 siblings: data not computed
Degree 44 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ 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.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
23Data not computed
47Data not computed