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Properties

Label 22.0.18130515624...7792.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{10}\cdot 7^{11}\cdot 11^{23}$
Root discriminant $44.47$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T31

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1129696, 161128, 2214300, 519860, 2198856, 434522, 1297219, 166903, 523688, -1177, 165286, -8698, 43703, -4796, 8855, -913, 1639, -77, 231, 0, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 22*x^20 + 231*x^18 - 77*x^17 + 1639*x^16 - 913*x^15 + 8855*x^14 - 4796*x^13 + 43703*x^12 - 8698*x^11 + 165286*x^10 - 1177*x^9 + 523688*x^8 + 166903*x^7 + 1297219*x^6 + 434522*x^5 + 2198856*x^4 + 519860*x^3 + 2214300*x^2 + 161128*x + 1129696)
 
gp: K = bnfinit(x^22 + 22*x^20 + 231*x^18 - 77*x^17 + 1639*x^16 - 913*x^15 + 8855*x^14 - 4796*x^13 + 43703*x^12 - 8698*x^11 + 165286*x^10 - 1177*x^9 + 523688*x^8 + 166903*x^7 + 1297219*x^6 + 434522*x^5 + 2198856*x^4 + 519860*x^3 + 2214300*x^2 + 161128*x + 1129696, 1)
 

Normalized defining polynomial

\( x^{22} + 22 x^{20} + 231 x^{18} - 77 x^{17} + 1639 x^{16} - 913 x^{15} + 8855 x^{14} - 4796 x^{13} + 43703 x^{12} - 8698 x^{11} + 165286 x^{10} - 1177 x^{9} + 523688 x^{8} + 166903 x^{7} + 1297219 x^{6} + 434522 x^{5} + 2198856 x^{4} + 519860 x^{3} + 2214300 x^{2} + 161128 x + 1129696 \)

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:  \(-1813051562475086060499058416925217792=-\,2^{10}\cdot 7^{11}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.47$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{8} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{18} + \frac{1}{16} a^{16} - \frac{1}{8} a^{15} - \frac{1}{4} a^{14} - \frac{5}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{5}{16} a^{8} - \frac{3}{8} a^{7} + \frac{5}{16} a^{6} - \frac{5}{16} a^{5} + \frac{5}{16} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{17} - \frac{1}{16} a^{14} + \frac{5}{16} a^{13} - \frac{3}{8} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{7}{16} a^{9} + \frac{1}{8} a^{8} + \frac{7}{16} a^{7} + \frac{1}{16} a^{6} - \frac{7}{16} a^{5} - \frac{3}{8} a^{3}$, $\frac{1}{32} a^{20} - \frac{1}{32} a^{18} - \frac{1}{8} a^{16} - \frac{1}{32} a^{15} - \frac{3}{32} a^{14} + \frac{5}{16} a^{13} + \frac{3}{8} a^{12} - \frac{3}{32} a^{10} - \frac{5}{16} a^{9} - \frac{13}{32} a^{8} - \frac{3}{32} a^{7} - \frac{7}{32} a^{6} + \frac{3}{8} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{79953268838804405227260121464471955637722236818429204389312} a^{21} - \frac{791730064755659027549883635312650652921059742127471233073}{79953268838804405227260121464471955637722236818429204389312} a^{20} + \frac{2398542705961041972948713147049578413985796055611791435157}{79953268838804405227260121464471955637722236818429204389312} a^{19} - \frac{1926082093150823749260454742526333900267419896105906249233}{79953268838804405227260121464471955637722236818429204389312} a^{18} + \frac{1143351461021479074388717152995976178563198878650270503521}{39976634419402202613630060732235977818861118409214602194656} a^{17} - \frac{9656258821958996787327555709722120740170652087607627203203}{79953268838804405227260121464471955637722236818429204389312} a^{16} + \frac{1630380274223740172707439511113821537977736869093724584365}{39976634419402202613630060732235977818861118409214602194656} a^{15} + \frac{2632207582082110658303266866187386110261701132509241877015}{79953268838804405227260121464471955637722236818429204389312} a^{14} + \frac{3626845384292395599637421011823108822140008965062856105185}{39976634419402202613630060732235977818861118409214602194656} a^{13} - \frac{2670623641983704070410606012240388187189987790847271286579}{39976634419402202613630060732235977818861118409214602194656} a^{12} - \frac{5398066255916612165344769886830018257767705107676434353339}{79953268838804405227260121464471955637722236818429204389312} a^{11} - \frac{3489846434159735011986861468169300898019080620957521286303}{79953268838804405227260121464471955637722236818429204389312} a^{10} - \frac{5911269512447719242016294573793358691252092363443777629861}{79953268838804405227260121464471955637722236818429204389312} a^{9} - \frac{4041336340666292307492973039223097671310477478967004717211}{19988317209701101306815030366117988909430559204607301097328} a^{8} + \frac{5645785734199628231425577880231875934615329562775296366479}{39976634419402202613630060732235977818861118409214602194656} a^{7} + \frac{13586280973150102060677328632265001589236892783362668280971}{79953268838804405227260121464471955637722236818429204389312} a^{6} + \frac{16820480516142430747749192307191269602862414116493177466169}{39976634419402202613630060732235977818861118409214602194656} a^{5} + \frac{990761153339001126646812412532649210747593949429361462107}{19988317209701101306815030366117988909430559204607301097328} a^{4} + \frac{3616716294432972799653197713433919177999565024730307892305}{9994158604850550653407515183058994454715279602303650548664} a^{3} + \frac{6797565533271860734642658770120539939797429430417416198661}{19988317209701101306815030366117988909430559204607301097328} a^{2} - \frac{2864065591913625335503245358885495519970498570751827766615}{9994158604850550653407515183058994454715279602303650548664} a + \frac{902305587687921906146368835503453268046218746412498870375}{2498539651212637663351878795764748613678819900575912637166}$

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:  \( 163923304373 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24200
The 77 conjugacy class representatives for t22n31 are not computed
Character table for t22n31 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \)

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

Sibling fields

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 R $20{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R $20{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.11.12.3$x^{11} + 77 x^{2} + 11$$11$$1$$12$$C_{11}:C_5$$[6/5]_{5}$
11.11.11.4$x^{11} + 77 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$