↑

Properties

Label 22.0.281...983.1
Degree $22$
Signature $[0, 11]$
Discriminant $-2.818\times 10^{24}$
Root discriminant $12.92$
Ramified prime $167$
Class number $1$
Class group trivial
Galois group $D_{11}$ (as 22T2)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 6*x^20 - 8*x^19 + 15*x^18 - 9*x^17 - 28*x^16 + 46*x^15 - 14*x^14 + 23*x^13 + 13*x^12 - 297*x^11 + 488*x^10 - 240*x^9 + 28*x^8 - 134*x^7 + 194*x^6 - 130*x^5 + 72*x^4 - 28*x^3 + 8*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^22 - 4*x^21 + 6*x^20 - 8*x^19 + 15*x^18 - 9*x^17 - 28*x^16 + 46*x^15 - 14*x^14 + 23*x^13 + 13*x^12 - 297*x^11 + 488*x^10 - 240*x^9 + 28*x^8 - 134*x^7 + 194*x^6 - 130*x^5 + 72*x^4 - 28*x^3 + 8*x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 8, -28, 72, -130, 194, -134, 28, -240, 488, -297, 13, 23, -14, 46, -28, -9, 15, -8, 6, -4, 1]);
 

\(x^{22} - 4 x^{21} + 6 x^{20} - 8 x^{19} + 15 x^{18} - 9 x^{17} - 28 x^{16} + 46 x^{15} - 14 x^{14} + 23 x^{13} + 13 x^{12} - 297 x^{11} + 488 x^{10} - 240 x^{9} + 28 x^{8} - 134 x^{7} + 194 x^{6} - 130 x^{5} + 72 x^{4} - 28 x^{3} + 8 x^{2} - 2 x + 1\)  Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-2817611963463158891460983\)\(\medspace = -\,167^{11}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.92$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $167$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $22$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{85} a^{19} + \frac{7}{85} a^{18} - \frac{2}{85} a^{17} + \frac{38}{85} a^{15} - \frac{8}{85} a^{14} - \frac{14}{85} a^{13} - \frac{22}{85} a^{12} + \frac{14}{85} a^{11} - \frac{1}{17} a^{10} - \frac{39}{85} a^{9} - \frac{2}{5} a^{8} - \frac{2}{85} a^{7} + \frac{13}{85} a^{6} + \frac{18}{85} a^{5} + \frac{21}{85} a^{4} + \frac{16}{85} a^{3} + \frac{8}{85} a^{2} + \frac{5}{17} a + \frac{13}{85}$, $\frac{1}{1105} a^{20} - \frac{1}{221} a^{19} + \frac{84}{1105} a^{18} + \frac{24}{1105} a^{17} - \frac{1}{85} a^{16} - \frac{532}{1105} a^{15} - \frac{89}{221} a^{14} + \frac{44}{1105} a^{13} + \frac{59}{221} a^{12} - \frac{479}{1105} a^{11} - \frac{268}{1105} a^{10} - \frac{3}{17} a^{9} + \frac{372}{1105} a^{8} - \frac{14}{1105} a^{7} - \frac{427}{1105} a^{6} - \frac{144}{1105} a^{5} - \frac{49}{1105} a^{4} - \frac{167}{1105} a^{3} - \frac{88}{1105} a^{2} + \frac{206}{1105} a - \frac{122}{1105}$, $\frac{1}{1029013899081155} a^{21} - \frac{138379852837}{1029013899081155} a^{20} - \frac{317079791929}{205802779816231} a^{19} - \frac{5731667921462}{1029013899081155} a^{18} + \frac{5341662730642}{79154915313935} a^{17} - \frac{2253417416977}{1029013899081155} a^{16} + \frac{188123120517446}{1029013899081155} a^{15} - \frac{18268489988497}{1029013899081155} a^{14} + \frac{287038856348799}{1029013899081155} a^{13} + \frac{14040616385044}{1029013899081155} a^{12} + \frac{228085414110377}{1029013899081155} a^{11} + \frac{7044091527234}{15830983062787} a^{10} + \frac{278340034354182}{1029013899081155} a^{9} + \frac{80417245071714}{205802779816231} a^{8} + \frac{377831056209817}{1029013899081155} a^{7} + \frac{40939162787522}{1029013899081155} a^{6} + \frac{346853154485672}{1029013899081155} a^{5} + \frac{20846897041610}{205802779816231} a^{4} - \frac{270886032588129}{1029013899081155} a^{3} + \frac{448462241310497}{1029013899081155} a^{2} + \frac{208494004079369}{1029013899081155} a - \frac{34349813749062}{79154915313935}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1029.39037079 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 1029.39037079 \cdot 1}{2\sqrt{2817611963463158891460983}}\approx 0.184751539673$

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-167}) \), 11.1.129891985607.1 x11

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

Sibling fields

Degree 11 sibling: 11.1.129891985607.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$167$167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$