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Properties

Label 22.0.49785181124...2411.2
Degree $22$
Signature $[0, 11]$
Discriminant $-\,11^{41}$
Root discriminant $87.25$
Ramified prime $11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{11}\times D_{11}$ (as 22T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76515613, 90503600, 96352960, 48915955, 12214961, -8569429, -9674137, -4704557, -198088, 1061170, 804166, 193023, -64306, -58828, -15686, 2706, 2673, 704, -66, -44, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^19 - 66*x^18 + 704*x^17 + 2673*x^16 + 2706*x^15 - 15686*x^14 - 58828*x^13 - 64306*x^12 + 193023*x^11 + 804166*x^10 + 1061170*x^9 - 198088*x^8 - 4704557*x^7 - 9674137*x^6 - 8569429*x^5 + 12214961*x^4 + 48915955*x^3 + 96352960*x^2 + 90503600*x + 76515613)
 
gp: K = bnfinit(x^22 - 44*x^19 - 66*x^18 + 704*x^17 + 2673*x^16 + 2706*x^15 - 15686*x^14 - 58828*x^13 - 64306*x^12 + 193023*x^11 + 804166*x^10 + 1061170*x^9 - 198088*x^8 - 4704557*x^7 - 9674137*x^6 - 8569429*x^5 + 12214961*x^4 + 48915955*x^3 + 96352960*x^2 + 90503600*x + 76515613, 1)
 

Normalized defining polynomial

\( x^{22} - 44 x^{19} - 66 x^{18} + 704 x^{17} + 2673 x^{16} + 2706 x^{15} - 15686 x^{14} - 58828 x^{13} - 64306 x^{12} + 193023 x^{11} + 804166 x^{10} + 1061170 x^{9} - 198088 x^{8} - 4704557 x^{7} - 9674137 x^{6} - 8569429 x^{5} + 12214961 x^{4} + 48915955 x^{3} + 96352960 x^{2} + 90503600 x + 76515613 \)

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:  \(-4978518112499354698647829163838661251242411=-\,11^{41}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.25$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{31} a^{20} - \frac{10}{31} a^{19} + \frac{15}{31} a^{18} + \frac{5}{31} a^{17} + \frac{4}{31} a^{16} - \frac{9}{31} a^{15} + \frac{5}{31} a^{14} + \frac{11}{31} a^{13} - \frac{8}{31} a^{12} - \frac{8}{31} a^{11} + \frac{4}{31} a^{10} + \frac{6}{31} a^{9} - \frac{2}{31} a^{8} + \frac{15}{31} a^{7} - \frac{9}{31} a^{6} - \frac{4}{31} a^{5} + \frac{1}{31} a^{4} + \frac{14}{31} a^{3} - \frac{8}{31} a^{2} + \frac{7}{31} a - \frac{10}{31}$, $\frac{1}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{21} - \frac{1040801415759676244192128485304305636475675462930127744997765004721607753809779548}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{20} - \frac{24476087781098970953657452083277732474343574523187212126620944644021298492047546406}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{19} + \frac{41114229145470672529206587045389364162716608196432097652742109083514394025420511637}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{18} - \frac{252728204245944529476942352548997162913900309863614785578077271405330700829997145}{1656078689849648892729319421389735956921921487285528544789406427751831597988294349} a^{17} - \frac{29671088210685600092828007224362106021396888627165601879319172906890380857411061399}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{16} + \frac{31881394136383757422566582170339807097276877429418523820837873028569896632939929997}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{15} - \frac{46026740707951386783246767215199646372288686649383903568848287470626637443413520017}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{14} - \frac{37091483669153639492921009628393925486876756376784991694579431629505847003108080995}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{13} - \frac{2186088250651998980182778140080238918512833040632584187200075252713407510214903035}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{12} + \frac{42784489248440608051823883184765358747247779894898353752018975570231449480615956265}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{11} - \frac{16208146655000510679654049248156770457204784609622067533850243708444691245963513584}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{10} - \frac{600281652975282255677984264078213456367645977004610071676287913670792625045188329}{3151891700036428537775156318128852305109463475801489811050805781850260138106753761} a^{9} - \frac{13898050378382835261279029971305680568593768676906061728711694825031306518485967390}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{8} - \frac{27901994254060337735709358731499370675806563002324423042634708526112571004023857149}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{7} - \frac{14337114648893132497492764526476692821551733087838571764116838061327964067364001138}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{6} + \frac{12484261139131902675766182677949151486523995873132779680488618190135838048756952311}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{5} - \frac{10658839418125269385354178919674908016619316529770807387486658065928222710051460412}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{4} - \frac{27895483493201057917610276685837253218340815981060705420371355506427821619360309443}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{3} + \frac{47177802224828550529897533098301766114188167999668899093369193206750379876984559421}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a^{2} - \frac{47948733224257983620647974534496630366766100443554633268227940999886391317014360486}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591} a - \frac{23572716663598219584697651956757882584072645679731893767552796533419517140260660606}{97708642701129284671029845861994421458393367749846184142574979237358064281309366591}$

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:  \( 1536390870260 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}\times D_{11}$ (as 22T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 242
The 77 conjugacy class representatives for $C_{11}\times D_{11}$ are not computed
Character table for $C_{11}\times D_{11}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \)

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 $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{11}$

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
11Data not computed