↑

Properties

Label 22.14.3790988231...9009.1
Degree $22$
Signature $[14, 4]$
Discriminant $23^{20}\cdot 47^{2}$
Root discriminant $24.54$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T23

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -18, 32, 76, -196, 17, 401, -487, -110, 554, -345, 234, -26, -469, 436, 15, -193, 68, 27, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 16*x^20 + 27*x^19 + 68*x^18 - 193*x^17 + 15*x^16 + 436*x^15 - 469*x^14 - 26*x^13 + 234*x^12 - 345*x^11 + 554*x^10 - 110*x^9 - 487*x^8 + 401*x^7 + 17*x^6 - 196*x^5 + 76*x^4 + 32*x^3 - 18*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^22 - x^21 - 16*x^20 + 27*x^19 + 68*x^18 - 193*x^17 + 15*x^16 + 436*x^15 - 469*x^14 - 26*x^13 + 234*x^12 - 345*x^11 + 554*x^10 - 110*x^9 - 487*x^8 + 401*x^7 + 17*x^6 - 196*x^5 + 76*x^4 + 32*x^3 - 18*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} - 16 x^{20} + 27 x^{19} + 68 x^{18} - 193 x^{17} + 15 x^{16} + 436 x^{15} - 469 x^{14} - 26 x^{13} + 234 x^{12} - 345 x^{11} + 554 x^{10} - 110 x^{9} - 487 x^{8} + 401 x^{7} + 17 x^{6} - 196 x^{5} + 76 x^{4} + 32 x^{3} - 18 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3790988231418101231518884499009=23^{20}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.54$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{776659267233153937} a^{21} + \frac{53785125729928735}{776659267233153937} a^{20} - \frac{45590071672339070}{776659267233153937} a^{19} - \frac{251596347616071496}{776659267233153937} a^{18} - \frac{57079011648014792}{776659267233153937} a^{17} + \frac{327226752907679799}{776659267233153937} a^{16} - \frac{197414525670064464}{776659267233153937} a^{15} + \frac{168254071732286767}{776659267233153937} a^{14} - \frac{222504280276825310}{776659267233153937} a^{13} - \frac{320914440620021865}{776659267233153937} a^{12} + \frac{210143878376135682}{776659267233153937} a^{11} - \frac{54083474102540801}{776659267233153937} a^{10} - \frac{274980363970388654}{776659267233153937} a^{9} + \frac{279578132111371269}{776659267233153937} a^{8} + \frac{385183874563872346}{776659267233153937} a^{7} + \frac{118341983875581885}{776659267233153937} a^{6} - \frac{296376415699345135}{776659267233153937} a^{5} + \frac{154093046679707075}{776659267233153937} a^{4} - \frac{365976824805321004}{776659267233153937} a^{3} + \frac{207837203191962076}{776659267233153937} a^{2} + \frac{9155938722476750}{776659267233153937} a - \frac{106777961271553306}{776659267233153937}$

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:  $17$
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:  \( 24016183.4425 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T23:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 11264
The 104 conjugacy class representatives for t22n23 are not computed
Character table for t22n23 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ 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
$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}$
47Data not computed