↑

Properties

Label 22.14.4098058278...8729.1
Degree $22$
Signature $[14, 4]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
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, -29, 289, -998, -659, 7657, -494, -20611, 2364, 24180, -7057, -16676, 6768, 6381, -2636, -652, 600, -173, -43, 51, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 43*x^18 - 173*x^17 + 600*x^16 - 652*x^15 - 2636*x^14 + 6381*x^13 + 6768*x^12 - 16676*x^11 - 7057*x^10 + 24180*x^9 + 2364*x^8 - 20611*x^7 - 494*x^6 + 7657*x^5 - 659*x^4 - 998*x^3 + 289*x^2 - 29*x + 1)
 
gp: K = bnfinit(x^22 - 4*x^21 - 7*x^20 + 51*x^19 - 43*x^18 - 173*x^17 + 600*x^16 - 652*x^15 - 2636*x^14 + 6381*x^13 + 6768*x^12 - 16676*x^11 - 7057*x^10 + 24180*x^9 + 2364*x^8 - 20611*x^7 - 494*x^6 + 7657*x^5 - 659*x^4 - 998*x^3 + 289*x^2 - 29*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 7 x^{20} + 51 x^{19} - 43 x^{18} - 173 x^{17} + 600 x^{16} - 652 x^{15} - 2636 x^{14} + 6381 x^{13} + 6768 x^{12} - 16676 x^{11} - 7057 x^{10} + 24180 x^{9} + 2364 x^{8} - 20611 x^{7} - 494 x^{6} + 7657 x^{5} - 659 x^{4} - 998 x^{3} + 289 x^{2} - 29 x + 1 \)

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:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{7958838244071398889574738886969} a^{21} + \frac{1030297339890685048743364449901}{7958838244071398889574738886969} a^{20} - \frac{3752923859272852641417741063831}{7958838244071398889574738886969} a^{19} - \frac{1473500020041627045486503365337}{7958838244071398889574738886969} a^{18} - \frac{1421060411996198564137817130429}{7958838244071398889574738886969} a^{17} + \frac{1501402590019436481497070446076}{7958838244071398889574738886969} a^{16} - \frac{900387824369122969361285525630}{7958838244071398889574738886969} a^{15} + \frac{316505085708941610130960249126}{7958838244071398889574738886969} a^{14} + \frac{2261585662572259538564649821396}{7958838244071398889574738886969} a^{13} + \frac{879287557472910402795051394408}{7958838244071398889574738886969} a^{12} + \frac{1197725854998987393098675644249}{7958838244071398889574738886969} a^{11} - \frac{3418135378215867710841247957895}{7958838244071398889574738886969} a^{10} - \frac{2966314290561710598565133894253}{7958838244071398889574738886969} a^{9} - \frac{3533612327077375758367532046460}{7958838244071398889574738886969} a^{8} - \frac{2196325649260142348873177628120}{7958838244071398889574738886969} a^{7} - \frac{2722588402135194771437802665378}{7958838244071398889574738886969} a^{6} + \frac{3381150499905972043738119733909}{7958838244071398889574738886969} a^{5} + \frac{3110865645356239373902099241963}{7958838244071398889574738886969} a^{4} - \frac{3899568605154244211499968112253}{7958838244071398889574738886969} a^{3} + \frac{2921304140235314704854072328946}{7958838244071398889574738886969} a^{2} + \frac{1480099753802824037476090895795}{7958838244071398889574738886969} a - \frac{2375603646617782813510744862618}{7958838244071398889574738886969}$

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:  \( 785042011.806 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
$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$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$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}$
47.2.1.2$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$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$