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Properties

Label 22.22.1157149888...5008.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{33}\cdot 1297^{10}$
Root discriminant $73.54$
Ramified primes $2, 1297$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{22}$ (as 22T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, -302, 4628, -17977, -24976, 287700, -501290, -51957, 777826, -385652, -383636, 328609, 51552, -105193, 11078, 14826, -3756, -782, 348, -5, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 10*x^21 - 5*x^20 + 348*x^19 - 782*x^18 - 3756*x^17 + 14826*x^16 + 11078*x^15 - 105193*x^14 + 51552*x^13 + 328609*x^12 - 383636*x^11 - 385652*x^10 + 777826*x^9 - 51957*x^8 - 501290*x^7 + 287700*x^6 - 24976*x^5 - 17977*x^4 + 4628*x^3 - 302*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^22 - 10*x^21 - 5*x^20 + 348*x^19 - 782*x^18 - 3756*x^17 + 14826*x^16 + 11078*x^15 - 105193*x^14 + 51552*x^13 + 328609*x^12 - 383636*x^11 - 385652*x^10 + 777826*x^9 - 51957*x^8 - 501290*x^7 + 287700*x^6 - 24976*x^5 - 17977*x^4 + 4628*x^3 - 302*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 10 x^{21} - 5 x^{20} + 348 x^{19} - 782 x^{18} - 3756 x^{17} + 14826 x^{16} + 11078 x^{15} - 105193 x^{14} + 51552 x^{13} + 328609 x^{12} - 383636 x^{11} - 385652 x^{10} + 777826 x^{9} - 51957 x^{8} - 501290 x^{7} + 287700 x^{6} - 24976 x^{5} - 17977 x^{4} + 4628 x^{3} - 302 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(115714988826933543891025354042178438955008=2^{33}\cdot 1297^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.54$
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:  $2, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{275} a^{19} - \frac{1}{55} a^{17} - \frac{1}{25} a^{16} - \frac{12}{275} a^{15} - \frac{21}{275} a^{14} - \frac{1}{11} a^{13} + \frac{16}{275} a^{12} + \frac{126}{275} a^{11} - \frac{73}{275} a^{10} - \frac{16}{55} a^{9} + \frac{26}{55} a^{8} - \frac{23}{55} a^{7} - \frac{64}{275} a^{6} + \frac{128}{275} a^{5} + \frac{1}{5} a^{4} - \frac{29}{275} a^{3} + \frac{27}{275} a^{2} + \frac{3}{275} a - \frac{21}{275}$, $\frac{1}{191675} a^{20} + \frac{14}{38335} a^{19} + \frac{648}{38335} a^{18} + \frac{16634}{191675} a^{17} - \frac{13322}{191675} a^{16} + \frac{16134}{191675} a^{15} - \frac{1113}{38335} a^{14} - \frac{1294}{191675} a^{13} + \frac{83911}{191675} a^{12} - \frac{82113}{191675} a^{11} - \frac{1522}{38335} a^{10} - \frac{43}{2255} a^{9} - \frac{11766}{38335} a^{8} - \frac{78074}{191675} a^{7} - \frac{36857}{191675} a^{6} - \frac{11958}{38335} a^{5} - \frac{28299}{191675} a^{4} + \frac{25442}{191675} a^{3} - \frac{73127}{191675} a^{2} - \frac{53161}{191675} a - \frac{12141}{38335}$, $\frac{1}{45456592283640092809160225} a^{21} - \frac{36742271334111070308}{45456592283640092809160225} a^{20} - \frac{35703765018476873519974}{45456592283640092809160225} a^{19} + \frac{179833041441085571421}{770450716671865979816275} a^{18} - \frac{18101636189476014646869}{4132417480330917528105475} a^{17} - \frac{4498571879910807361193961}{45456592283640092809160225} a^{16} - \frac{4063789559628531142642829}{45456592283640092809160225} a^{15} + \frac{99542959143133738393363}{1818263691345603712366409} a^{14} + \frac{87568464731687536382003}{45456592283640092809160225} a^{13} + \frac{2092290893552270633547134}{9091318456728018561832045} a^{12} - \frac{3545785418325054241088562}{9091318456728018561832045} a^{11} - \frac{727343325190792904050208}{4132417480330917528105475} a^{10} + \frac{18138639182493666265097}{44347894910868383228449} a^{9} + \frac{5230923503662899649746886}{45456592283640092809160225} a^{8} + \frac{3785486132080606029960662}{9091318456728018561832045} a^{7} + \frac{17670871082356435117306777}{45456592283640092809160225} a^{6} - \frac{3832877568996026938422391}{45456592283640092809160225} a^{5} + \frac{20896569443791990871184594}{45456592283640092809160225} a^{4} - \frac{19697459175380793148029}{45320630392462704695075} a^{3} + \frac{11214102184417388069232447}{45456592283640092809160225} a^{2} - \frac{1101076534835264170383107}{2673917193155299577009425} a + \frac{5590958645482105394879294}{45456592283640092809160225}$

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:  $21$
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:  \( 40553964823400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{22}$ (as 22T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 44
The 14 conjugacy class representatives for $D_{22}$
Character table for $D_{22}$

Intermediate fields

\(\Q(\sqrt{2}) \), 11.11.3670285774226257.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 22 sibling: 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 R $22$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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
2Data not computed
1297Data not computed