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Properties

Label 22.0.19194342495...1811.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,11^{31}$
Root discriminant $29.34$
Ramified prime $11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{11}$ (as 22T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -2816, 14960, -52624, 132748, -233860, 271315, -188375, 48642, 49665, -77803, 60224, -30690, 9482, 1100, -4070, 3454, -1980, 847, -275, 66, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 66*x^20 - 275*x^19 + 847*x^18 - 1980*x^17 + 3454*x^16 - 4070*x^15 + 1100*x^14 + 9482*x^13 - 30690*x^12 + 60224*x^11 - 77803*x^10 + 49665*x^9 + 48642*x^8 - 188375*x^7 + 271315*x^6 - 233860*x^5 + 132748*x^4 - 52624*x^3 + 14960*x^2 - 2816*x + 256)
 
gp: K = bnfinit(x^22 - 11*x^21 + 66*x^20 - 275*x^19 + 847*x^18 - 1980*x^17 + 3454*x^16 - 4070*x^15 + 1100*x^14 + 9482*x^13 - 30690*x^12 + 60224*x^11 - 77803*x^10 + 49665*x^9 + 48642*x^8 - 188375*x^7 + 271315*x^6 - 233860*x^5 + 132748*x^4 - 52624*x^3 + 14960*x^2 - 2816*x + 256, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 66 x^{20} - 275 x^{19} + 847 x^{18} - 1980 x^{17} + 3454 x^{16} - 4070 x^{15} + 1100 x^{14} + 9482 x^{13} - 30690 x^{12} + 60224 x^{11} - 77803 x^{10} + 49665 x^{9} + 48642 x^{8} - 188375 x^{7} + 271315 x^{6} - 233860 x^{5} + 132748 x^{4} - 52624 x^{3} + 14960 x^{2} - 2816 x + 256 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-191943424957750480504146841291811=-\,11^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.34$
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:  $11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{2176} a^{18} - \frac{9}{2176} a^{17} + \frac{1}{136} a^{16} - \frac{15}{544} a^{15} - \frac{7}{544} a^{14} + \frac{7}{272} a^{13} - \frac{43}{1088} a^{12} - \frac{43}{1088} a^{11} - \frac{27}{272} a^{10} - \frac{1}{8} a^{9} + \frac{27}{544} a^{8} - \frac{27}{272} a^{7} + \frac{293}{2176} a^{6} + \frac{399}{2176} a^{5} + \frac{67}{272} a^{4} + \frac{15}{32} a^{3} - \frac{7}{68} a^{2} + \frac{59}{136} a + \frac{15}{34}$, $\frac{1}{2176} a^{19} + \frac{3}{2176} a^{17} + \frac{1}{136} a^{16} - \frac{3}{272} a^{15} - \frac{15}{544} a^{14} + \frac{5}{1088} a^{13} + \frac{23}{544} a^{12} - \frac{87}{1088} a^{11} + \frac{29}{272} a^{10} - \frac{7}{544} a^{9} - \frac{49}{544} a^{8} + \frac{389}{2176} a^{7} + \frac{113}{544} a^{6} + \frac{523}{2176} a^{5} - \frac{43}{272} a^{4} - \frac{243}{544} a^{3} + \frac{9}{68} a^{2} + \frac{47}{136} a - \frac{1}{34}$, $\frac{1}{8742993761152} a^{20} - \frac{5}{4371496880576} a^{19} + \frac{1804607407}{8742993761152} a^{18} - \frac{8120733189}{4371496880576} a^{17} + \frac{27641910395}{2185748440288} a^{16} + \frac{7508970177}{2185748440288} a^{15} + \frac{12895353741}{4371496880576} a^{14} - \frac{1283166235}{136609277518} a^{13} - \frac{154397649059}{4371496880576} a^{12} - \frac{122458005907}{2185748440288} a^{11} - \frac{200850141955}{2185748440288} a^{10} - \frac{111402170293}{2185748440288} a^{9} + \frac{345409205765}{8742993761152} a^{8} + \frac{1017937710869}{4371496880576} a^{7} - \frac{1605350467689}{8742993761152} a^{6} - \frac{600150347253}{4371496880576} a^{5} - \frac{2922338095}{136609277518} a^{4} - \frac{77662894039}{546437110072} a^{3} + \frac{50441702895}{136609277518} a^{2} + \frac{18860479097}{273218555036} a - \frac{188576621}{68304638759}$, $\frac{1}{3086276797686656} a^{21} + \frac{83}{1543138398843328} a^{20} + \frac{91305501181}{1543138398843328} a^{19} - \frac{707128464423}{3086276797686656} a^{18} - \frac{2026730266811}{220448342691904} a^{17} + \frac{2798168311217}{192892299855416} a^{16} - \frac{37563325613225}{1543138398843328} a^{15} + \frac{116899289997}{55112085672976} a^{14} - \frac{412413200531}{13778021418244} a^{13} - \frac{32565857672685}{1543138398843328} a^{12} + \frac{265677085587}{13778021418244} a^{11} + \frac{35905266589001}{771569199421664} a^{10} + \frac{180238389266177}{3086276797686656} a^{9} + \frac{7322499935897}{1543138398843328} a^{8} - \frac{196129838037045}{1543138398843328} a^{7} + \frac{2725496106423}{25935099140224} a^{6} + \frac{243497534519637}{1543138398843328} a^{5} + \frac{52038223087703}{771569199421664} a^{4} - \frac{13049912658891}{48223074963854} a^{3} - \frac{19931360727339}{48223074963854} a^{2} + \frac{22958338540321}{48223074963854} a + \frac{4607508030389}{24111537481927}$

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:  $10$
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:  \( 167976253.311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{11}$ (as 22T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22
The 7 conjugacy class representatives for $D_{11}$
Character table for $D_{11}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 11.1.4177248169415651.1 x11

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

Sibling fields

Degree 11 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{11}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
11Data not computed