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Properties

Label 22.22.7639732423...3125.1
Degree $22$
Signature $[22, 0]$
Discriminant $3^{20}\cdot 5^{21}\cdot 11^{16}$
Root discriminant $72.16$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{11}:C_5$ (as 22T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -9, -3964, 13185, 46165, -122958, -216132, 326688, 414510, -406980, -379689, 285909, 180264, -120705, -44775, 30348, 5307, -4323, -170, 315, -16, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 16*x^20 + 315*x^19 - 170*x^18 - 4323*x^17 + 5307*x^16 + 30348*x^15 - 44775*x^14 - 120705*x^13 + 180264*x^12 + 285909*x^11 - 379689*x^10 - 406980*x^9 + 414510*x^8 + 326688*x^7 - 216132*x^6 - 122958*x^5 + 46165*x^4 + 13185*x^3 - 3964*x^2 - 9*x + 19)
 
gp: K = bnfinit(x^22 - 9*x^21 - 16*x^20 + 315*x^19 - 170*x^18 - 4323*x^17 + 5307*x^16 + 30348*x^15 - 44775*x^14 - 120705*x^13 + 180264*x^12 + 285909*x^11 - 379689*x^10 - 406980*x^9 + 414510*x^8 + 326688*x^7 - 216132*x^6 - 122958*x^5 + 46165*x^4 + 13185*x^3 - 3964*x^2 - 9*x + 19, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} - 16 x^{20} + 315 x^{19} - 170 x^{18} - 4323 x^{17} + 5307 x^{16} + 30348 x^{15} - 44775 x^{14} - 120705 x^{13} + 180264 x^{12} + 285909 x^{11} - 379689 x^{10} - 406980 x^{9} + 414510 x^{8} + 326688 x^{7} - 216132 x^{6} - 122958 x^{5} + 46165 x^{4} + 13185 x^{3} - 3964 x^{2} - 9 x + 19 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(76397324237092623287515907764434814453125=3^{20}\cdot 5^{21}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.16$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{10} - \frac{2}{5} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{10} - \frac{2}{5} a^{9} - \frac{1}{2} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{11} - \frac{2}{5} a^{10} - \frac{1}{2} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a$, $\frac{1}{10} a^{16} + \frac{3}{10} a^{10} - \frac{1}{2} a^{9} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{11} + \frac{3}{10} a^{10} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{50} a^{18} - \frac{1}{50} a^{17} + \frac{1}{25} a^{16} + \frac{1}{25} a^{15} - \frac{1}{50} a^{13} + \frac{1}{50} a^{12} + \frac{3}{50} a^{11} - \frac{17}{50} a^{10} + \frac{1}{5} a^{9} - \frac{4}{25} a^{8} - \frac{6}{25} a^{7} + \frac{12}{25} a^{6} - \frac{3}{25} a^{5} + \frac{1}{10} a^{4} - \frac{13}{50} a^{3} + \frac{3}{50} a^{2} + \frac{12}{25} a + \frac{7}{25}$, $\frac{1}{50} a^{19} + \frac{1}{50} a^{17} - \frac{1}{50} a^{16} + \frac{1}{25} a^{15} - \frac{1}{50} a^{14} - \frac{1}{50} a^{12} - \frac{2}{25} a^{11} + \frac{3}{50} a^{10} - \frac{23}{50} a^{9} - \frac{2}{5} a^{8} - \frac{9}{25} a^{7} - \frac{6}{25} a^{6} - \frac{8}{25} a^{5} + \frac{17}{50} a^{4} - \frac{1}{5} a^{3} - \frac{9}{25} a^{2} - \frac{1}{25} a + \frac{2}{25}$, $\frac{1}{473050} a^{20} + \frac{1127}{236525} a^{19} - \frac{1749}{473050} a^{18} - \frac{11256}{236525} a^{17} + \frac{1069}{236525} a^{16} + \frac{7301}{236525} a^{15} - \frac{12739}{473050} a^{14} - \frac{4361}{473050} a^{13} - \frac{19143}{473050} a^{12} + \frac{19187}{473050} a^{11} - \frac{81248}{236525} a^{10} + \frac{65289}{236525} a^{9} - \frac{122833}{473050} a^{8} + \frac{58861}{473050} a^{7} + \frac{20053}{236525} a^{6} - \frac{114697}{473050} a^{5} + \frac{70234}{236525} a^{4} - \frac{55824}{236525} a^{3} - \frac{142729}{473050} a^{2} + \frac{71618}{236525} a + \frac{7023}{236525}$, $\frac{1}{90069438793886561914250588050} a^{21} - \frac{56442580445124338694751}{90069438793886561914250588050} a^{20} + \frac{158835199904026076675381876}{45034719396943280957125294025} a^{19} + \frac{313339493365787069367471016}{45034719396943280957125294025} a^{18} - \frac{327786805972950637165882303}{18013887758777312382850117610} a^{17} - \frac{2000134801723720153534954151}{90069438793886561914250588050} a^{16} - \frac{550181657292479321826347829}{90069438793886561914250588050} a^{15} + \frac{1977056445334757225286217354}{45034719396943280957125294025} a^{14} - \frac{627646957437415692387832807}{90069438793886561914250588050} a^{13} - \frac{55428447740591434807106991}{3602777551755462476570023522} a^{12} - \frac{7718098406808582015498241103}{90069438793886561914250588050} a^{11} - \frac{14940988324511090755185114767}{90069438793886561914250588050} a^{10} - \frac{14846565908677925651308164548}{45034719396943280957125294025} a^{9} - \frac{14805564910851563564977413103}{45034719396943280957125294025} a^{8} + \frac{754748017371176690670783700}{1801388775877731238285011761} a^{7} + \frac{17709935313945382366636838367}{90069438793886561914250588050} a^{6} - \frac{35240477886091270283048339317}{90069438793886561914250588050} a^{5} - \frac{1021646355947228690082841231}{90069438793886561914250588050} a^{4} - \frac{22838917109594767974459522841}{90069438793886561914250588050} a^{3} - \frac{656083998734101944783203451}{18013887758777312382850117610} a^{2} - \frac{6888596119882282487697853771}{18013887758777312382850117610} a - \frac{427181737442356944791676037}{1801388775877731238285011761}$

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:  $21$
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:  \( 73548357254400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{11}:C_5$ (as 22T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 14 conjugacy class representatives for $C_2\times C_{11}:C_5$
Character table for $C_2\times C_{11}:C_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 11.11.123610132462587890625.1

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

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$