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Properties

Label 22.6.11342052492...3125.1
Degree $22$
Signature $[6, 8]$
Discriminant $5^{8}\cdot 7^{15}\cdot 11^{19}$
Root discriminant $53.67$
Ramified primes $5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T14

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6300, -17850, 53515, 37800, -239399, 244344, 84439, -422621, 431163, -188822, -19458, 76979, -51771, 22033, -7373, 1506, 530, -720, 383, -140, 41, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300)
 
gp: K = bnfinit(x^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} + 41 x^{20} - 140 x^{19} + 383 x^{18} - 720 x^{17} + 530 x^{16} + 1506 x^{15} - 7373 x^{14} + 22033 x^{13} - 51771 x^{12} + 76979 x^{11} - 19458 x^{10} - 188822 x^{9} + 431163 x^{8} - 422621 x^{7} + 84439 x^{6} + 244344 x^{5} - 239399 x^{4} + 37800 x^{3} + 53515 x^{2} - 17850 x - 6300 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(113420524920318543882697355574770703125=5^{8}\cdot 7^{15}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.67$
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:  $5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{14} a^{17} + \frac{1}{14} a^{16} - \frac{2}{7} a^{15} - \frac{3}{14} a^{14} - \frac{1}{2} a^{13} + \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{8} - \frac{1}{14} a^{7} + \frac{3}{14} a^{6} + \frac{5}{14} a^{5} - \frac{1}{14} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{18} + \frac{1}{7} a^{16} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{2} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{5}{14} a^{8} + \frac{2}{7} a^{7} - \frac{5}{14} a^{6} + \frac{1}{14} a^{5} - \frac{3}{7} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{19} - \frac{1}{14} a^{16} - \frac{3}{14} a^{15} - \frac{3}{7} a^{14} - \frac{1}{14} a^{12} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{5}{14} a^{9} - \frac{2}{7} a^{8} - \frac{3}{14} a^{7} + \frac{1}{7} a^{6} + \frac{5}{14} a^{5} - \frac{5}{14} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2030} a^{20} - \frac{1}{70} a^{19} + \frac{1}{2030} a^{18} - \frac{3}{406} a^{17} + \frac{9}{290} a^{16} - \frac{80}{203} a^{15} + \frac{38}{203} a^{14} + \frac{193}{1015} a^{13} + \frac{447}{2030} a^{12} + \frac{59}{1015} a^{11} + \frac{199}{2030} a^{10} + \frac{999}{2030} a^{9} - \frac{499}{1015} a^{8} - \frac{747}{2030} a^{7} - \frac{597}{2030} a^{6} + \frac{202}{1015} a^{5} - \frac{123}{1015} a^{4} - \frac{4}{145} a^{3} + \frac{53}{290} a^{2} - \frac{25}{58} a + \frac{5}{29}$, $\frac{1}{100273398988248821503734020890696199190} a^{21} - \frac{2552519842161323003804894660024118}{16712233164708136917289003481782699865} a^{20} + \frac{31809203014824593634169253042123633}{2046395897719363704157837161034616310} a^{19} + \frac{861352122178327957238335957946084}{7162385642017772964552430063621157085} a^{18} - \frac{1229452367825127930477660986083908431}{50136699494124410751867010445348099595} a^{17} - \frac{187588623866856645400236443917736696}{2387461880672590988184143354540385695} a^{16} - \frac{9590146591362823884059648128741868597}{20054679797649764300746804178139239838} a^{15} + \frac{306956005911694215096469009079830023}{2387461880672590988184143354540385695} a^{14} + \frac{20937700971397471226801889472420696063}{100273398988248821503734020890696199190} a^{13} + \frac{3097338068858585298924105066931922956}{10027339898824882150373402089069619919} a^{12} - \frac{293701935350281464499093207989542991}{33424466329416273834578006963565399730} a^{11} - \frac{16807994191156353327651357904852796351}{50136699494124410751867010445348099595} a^{10} + \frac{305396793460282845412039938535108013}{2387461880672590988184143354540385695} a^{9} - \frac{194602871837503617116809105334591789}{691540682677578079336096695797904822} a^{8} + \frac{16173879148891582874747731933476567257}{33424466329416273834578006963565399730} a^{7} + \frac{43247799040214369953523575134234493117}{100273398988248821503734020890696199190} a^{6} - \frac{12759257769368425284114263717497257647}{100273398988248821503734020890696199190} a^{5} - \frac{5121232304675559883114646156868084629}{33424466329416273834578006963565399730} a^{4} + \frac{4444515736462521280088130722908327}{14113075156685266925226463179549078} a^{3} - \frac{1037407775666746636231206796350925739}{4774923761345181976368286709080771390} a^{2} + \frac{666844606866346412060794316434200631}{1432477128403554592910486012724231417} a + \frac{140281478732381844197184661354017708}{477492376134518197636828670908077139}$

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:  $13$
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:  \( 118924316463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T14:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 13 conjugacy class representatives for t22n14
Character table for t22n14

Intermediate fields

\(\Q(\sqrt{77}) \)

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.6.5.2$x^{6} + 33$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
11.12.11.1$x^{12} + 33$$12$$1$$11$$D_{12}$$[\ ]_{12}^{2}$