↑

Properties

Label 22.0.22085756220...6963.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{21}\cdot 11^{32}$
Root discriminant $93.37$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T11

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3159, -18711, 29403, 53460, -147015, -31185, 313632, 3564, -343035, 594, 222156, -81, -91179, 0, 24552, 0, -4356, 0, 495, 0, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 33*x^20 + 495*x^18 - 4356*x^16 + 24552*x^14 - 91179*x^12 - 81*x^11 + 222156*x^10 + 594*x^9 - 343035*x^8 + 3564*x^7 + 313632*x^6 - 31185*x^5 - 147015*x^4 + 53460*x^3 + 29403*x^2 - 18711*x + 3159)
 
gp: K = bnfinit(x^22 - 33*x^20 + 495*x^18 - 4356*x^16 + 24552*x^14 - 91179*x^12 - 81*x^11 + 222156*x^10 + 594*x^9 - 343035*x^8 + 3564*x^7 + 313632*x^6 - 31185*x^5 - 147015*x^4 + 53460*x^3 + 29403*x^2 - 18711*x + 3159, 1)
 

Normalized defining polynomial

\( x^{22} - 33 x^{20} + 495 x^{18} - 4356 x^{16} + 24552 x^{14} - 91179 x^{12} - 81 x^{11} + 222156 x^{10} + 594 x^{9} - 343035 x^{8} + 3564 x^{7} + 313632 x^{6} - 31185 x^{5} - 147015 x^{4} + 53460 x^{3} + 29403 x^{2} - 18711 x + 3159 \)

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:  \(-22085756220567549162579756126980406366726963=-\,3^{21}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.37$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{9} a^{9}$, $\frac{1}{9} a^{10}$, $\frac{1}{9} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{9} a^{13}$, $\frac{1}{27} a^{14}$, $\frac{1}{27} a^{15}$, $\frac{1}{27} a^{16}$, $\frac{1}{27} a^{17}$, $\frac{1}{81} a^{18}$, $\frac{1}{81} a^{19}$, $\frac{1}{81} a^{20}$, $\frac{1}{156451963351005060464493} a^{21} + \frac{168052396502361954646}{52150654450335020154831} a^{20} + \frac{21460319100705919028}{17383551483445006718277} a^{19} - \frac{901173334592346071969}{156451963351005060464493} a^{18} + \frac{28944416186079124534}{1931505720382778524253} a^{17} - \frac{471153292212590915944}{52150654450335020154831} a^{16} - \frac{789592978438162538050}{52150654450335020154831} a^{15} + \frac{307364714005453827155}{17383551483445006718277} a^{14} - \frac{252069800821512008017}{17383551483445006718277} a^{13} + \frac{646868872593982736377}{17383551483445006718277} a^{12} - \frac{86274914943003834242}{5794517161148335572759} a^{11} + \frac{83643656470652328624}{1931505720382778524253} a^{10} + \frac{580425510413436350594}{17383551483445006718277} a^{9} + \frac{457936260956091980413}{5794517161148335572759} a^{8} - \frac{594647944519182296018}{5794517161148335572759} a^{7} + \frac{609631561065522611723}{5794517161148335572759} a^{6} - \frac{155747212009204773685}{5794517161148335572759} a^{5} + \frac{538793892321863763344}{1931505720382778524253} a^{4} - \frac{496352484826650965382}{1931505720382778524253} a^{3} - \frac{595689762486098368059}{1931505720382778524253} a^{2} + \frac{823253418188365075591}{1931505720382778524253} a + \frac{488985030338658626953}{1931505720382778524253}$

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:  \( \frac{90984746206135690205}{156451963351005060464493} a^{21} - \frac{43951915608327660791}{156451963351005060464493} a^{20} - \frac{987210760935218223118}{52150654450335020154831} a^{19} + \frac{464169520173848216167}{52150654450335020154831} a^{18} + \frac{14575402557152001555287}{52150654450335020154831} a^{17} - \frac{6638934446893561365410}{52150654450335020154831} a^{16} - \frac{41922222885949034730244}{17383551483445006718277} a^{15} + \frac{6112165022214725543036}{5794517161148335572759} a^{14} + \frac{230362895549622040663106}{17383551483445006718277} a^{13} - \frac{31797541355715811399306}{5794517161148335572759} a^{12} - \frac{826631974996098426653189}{17383551483445006718277} a^{11} + \frac{35041454194892104895441}{1931505720382778524253} a^{10} + \frac{213268430563113148223737}{1931505720382778524253} a^{9} - \frac{213963942865101724978870}{5794517161148335572759} a^{8} - \frac{306727206888702323079070}{1931505720382778524253} a^{7} + \frac{86519951876596013136515}{1931505720382778524253} a^{6} + \frac{248708063814910177221920}{1931505720382778524253} a^{5} - \frac{72155603263624402456715}{1931505720382778524253} a^{4} - \frac{83786054891972123382362}{1931505720382778524253} a^{3} + \frac{40362501073728799595553}{1931505720382778524253} a^{2} + \frac{6139844254675023638889}{1931505720382778524253} a - \frac{2458820957519800985333}{1931505720382778524253} \) (order $6$)
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:  \( 25313714393700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T11:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1210
The 25 conjugacy class representatives for t22n11
Character table for t22n11 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
11Data not computed