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Properties

Label 22.10.4098058278...8729.3
Degree $22$
Signature $[10, 6]$
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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 34, 121, -1429, -1252, 7273, 10658, 8908, 8081, 4895, 1428, 68, 196, -650, -135, -59, -166, 14, -30, -8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 4*x^20 - 8*x^19 - 30*x^18 + 14*x^17 - 166*x^16 - 59*x^15 - 135*x^14 - 650*x^13 + 196*x^12 + 68*x^11 + 1428*x^10 + 4895*x^9 + 8081*x^8 + 8908*x^7 + 10658*x^6 + 7273*x^5 - 1252*x^4 - 1429*x^3 + 121*x^2 + 34*x + 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + 4*x^20 - 8*x^19 - 30*x^18 + 14*x^17 - 166*x^16 - 59*x^15 - 135*x^14 - 650*x^13 + 196*x^12 + 68*x^11 + 1428*x^10 + 4895*x^9 + 8081*x^8 + 8908*x^7 + 10658*x^6 + 7273*x^5 - 1252*x^4 - 1429*x^3 + 121*x^2 + 34*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 4 x^{20} - 8 x^{19} - 30 x^{18} + 14 x^{17} - 166 x^{16} - 59 x^{15} - 135 x^{14} - 650 x^{13} + 196 x^{12} + 68 x^{11} + 1428 x^{10} + 4895 x^{9} + 8081 x^{8} + 8908 x^{7} + 10658 x^{6} + 7273 x^{5} - 1252 x^{4} - 1429 x^{3} + 121 x^{2} + 34 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1637232890027778397248060015822326437} a^{21} + \frac{724991019353271198730792206745059262}{1637232890027778397248060015822326437} a^{20} + \frac{36545593412976303636492793498716299}{1637232890027778397248060015822326437} a^{19} + \frac{180511612867004294312907742181551699}{1637232890027778397248060015822326437} a^{18} + \frac{73089427842403552780103209433117214}{1637232890027778397248060015822326437} a^{17} - \frac{521352332951396914437498692298911217}{1637232890027778397248060015822326437} a^{16} - \frac{817647621779391850817688043442246271}{1637232890027778397248060015822326437} a^{15} + \frac{472459112619554499365978392972271095}{1637232890027778397248060015822326437} a^{14} - \frac{628111815937909822017910588583900503}{1637232890027778397248060015822326437} a^{13} - \frac{551749925336438084943205777659574753}{1637232890027778397248060015822326437} a^{12} + \frac{757719883931484700675576748458876651}{1637232890027778397248060015822326437} a^{11} + \frac{302830742293039414623263058831697078}{1637232890027778397248060015822326437} a^{10} - \frac{373573504885228976547160468765271305}{1637232890027778397248060015822326437} a^{9} - \frac{534546534427229846719780922543608110}{1637232890027778397248060015822326437} a^{8} + \frac{228727955142170218712076463141407159}{1637232890027778397248060015822326437} a^{7} + \frac{456248953913382074103610335771088008}{1637232890027778397248060015822326437} a^{6} + \frac{495450864526017244145897618835228450}{1637232890027778397248060015822326437} a^{5} - \frac{794259444045651153304028114177182556}{1637232890027778397248060015822326437} a^{4} + \frac{797957213422823788934197986446976328}{1637232890027778397248060015822326437} a^{3} + \frac{227939605266727629277372511890602409}{1637232890027778397248060015822326437} a^{2} + \frac{99241846648447428792371986493140733}{1637232890027778397248060015822326437} a + \frac{752751170906942346706615527962567579}{1637232890027778397248060015822326437}$

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:  $15$
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:  \( 297447714.815 \) (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
47Data not computed