↑

Properties

Label 22.0.38582011381...3536.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{10}\cdot 7^{11}\cdot 11^{18}\cdot 17^{11}$
Root discriminant $106.33$
Ramified primes $2, 7, 11, 17$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group 22T25

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![197813583, -209238282, 174513339, -344473206, 522042400, -543667399, 422694019, -264596266, 143941842, -71190955, 32472704, -13519264, 5129410, -1807377, 587862, -172722, 44781, -10043, 2079, -396, 58, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 58*x^20 - 396*x^19 + 2079*x^18 - 10043*x^17 + 44781*x^16 - 172722*x^15 + 587862*x^14 - 1807377*x^13 + 5129410*x^12 - 13519264*x^11 + 32472704*x^10 - 71190955*x^9 + 143941842*x^8 - 264596266*x^7 + 422694019*x^6 - 543667399*x^5 + 522042400*x^4 - 344473206*x^3 + 174513339*x^2 - 209238282*x + 197813583)
 
gp: K = bnfinit(x^22 - 8*x^21 + 58*x^20 - 396*x^19 + 2079*x^18 - 10043*x^17 + 44781*x^16 - 172722*x^15 + 587862*x^14 - 1807377*x^13 + 5129410*x^12 - 13519264*x^11 + 32472704*x^10 - 71190955*x^9 + 143941842*x^8 - 264596266*x^7 + 422694019*x^6 - 543667399*x^5 + 522042400*x^4 - 344473206*x^3 + 174513339*x^2 - 209238282*x + 197813583, 1)
 

Normalized defining polynomial

\( x^{22} - 8 x^{21} + 58 x^{20} - 396 x^{19} + 2079 x^{18} - 10043 x^{17} + 44781 x^{16} - 172722 x^{15} + 587862 x^{14} - 1807377 x^{13} + 5129410 x^{12} - 13519264 x^{11} + 32472704 x^{10} - 71190955 x^{9} + 143941842 x^{8} - 264596266 x^{7} + 422694019 x^{6} - 543667399 x^{5} + 522042400 x^{4} - 344473206 x^{3} + 174513339 x^{2} - 209238282 x + 197813583 \)

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:  \(-385820113812010750478864385892000309956033536=-\,2^{10}\cdot 7^{11}\cdot 11^{18}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.33$
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:  $2, 7, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{3}{11} a^{11} + \frac{3}{11} a^{10}$, $\frac{1}{11} a^{13} + \frac{5}{11} a^{11} + \frac{2}{11} a^{10}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{11} - \frac{4}{11} a^{10}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{11} - \frac{5}{11} a^{10}$, $\frac{1}{33} a^{16} + \frac{1}{33} a^{15} + \frac{1}{33} a^{12} + \frac{16}{33} a^{11} + \frac{14}{33} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{33} a^{17} - \frac{1}{33} a^{15} + \frac{1}{33} a^{13} - \frac{14}{33} a^{11} + \frac{7}{33} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{33} a^{18} + \frac{1}{33} a^{15} + \frac{1}{33} a^{14} - \frac{1}{33} a^{12} - \frac{7}{33} a^{11} - \frac{5}{33} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{165} a^{19} + \frac{1}{165} a^{16} + \frac{7}{165} a^{15} + \frac{2}{165} a^{13} - \frac{7}{165} a^{12} + \frac{1}{3} a^{11} - \frac{19}{55} a^{10} - \frac{1}{3} a^{9} + \frac{7}{15} a^{8} + \frac{1}{15} a^{7} + \frac{7}{15} a^{6} - \frac{2}{5} a^{5} + \frac{2}{15} a^{4} + \frac{2}{15} a^{3} + \frac{4}{15} a^{2} + \frac{2}{5}$, $\frac{1}{495} a^{20} + \frac{1}{495} a^{19} - \frac{1}{99} a^{18} + \frac{2}{165} a^{17} + \frac{1}{165} a^{16} + \frac{7}{495} a^{15} - \frac{1}{165} a^{14} - \frac{1}{33} a^{13} + \frac{1}{165} a^{12} - \frac{4}{15} a^{11} + \frac{133}{495} a^{10} + \frac{22}{45} a^{9} + \frac{13}{45} a^{8} - \frac{2}{45} a^{7} + \frac{7}{15} a^{6} + \frac{16}{45} a^{5} - \frac{1}{45} a^{4} + \frac{16}{45} a^{3} - \frac{16}{45} a^{2} + \frac{7}{15} a - \frac{1}{5}$, $\frac{1}{26065376228352945822594585948143912919380628966221624735468858617239075580410} a^{21} + \frac{6801452406085420322462525205708086855106664530055735447274915657412119293}{26065376228352945822594585948143912919380628966221624735468858617239075580410} a^{20} - \frac{25754874777690930841366875541458552746484321416449320182075124884356774709}{26065376228352945822594585948143912919380628966221624735468858617239075580410} a^{19} + \frac{1971621130956425493176367082798784202603034768916633389144038148550168029}{263286628569221674975702888365090029488693221881026512479483420376152278590} a^{18} - \frac{158910005607944769315801488065553094160509724753922443231132102217006420}{26328662856922167497570288836509002948869322188102651247948342037615227859} a^{17} + \frac{2753385431881500829701087257121549730126941601732219734050984494040794367}{2369579657122995074781325995285810265398238996929238612315350783385370507310} a^{16} - \frac{7769508657623671926733658532635404692623284315352076430319822157055881796}{394929942853832512463554332547635044233039832821539768719225130564228417885} a^{15} - \frac{13173449946139221583214609954701505942928095926132230068194504044121173226}{394929942853832512463554332547635044233039832821539768719225130564228417885} a^{14} - \frac{1743678212872273230359757679368658902036197915053864614894872423828975246}{43881104761536945829283814727515004914782203646837752079913903396025379765} a^{13} + \frac{638860670061199162127253784114214766177296156244646384821284397763156877}{789859885707665024927108665095270088466079665643079537438450261128456835770} a^{12} - \frac{289302381250611122181008874333676757321455064445223503407578063234193110681}{2369579657122995074781325995285810265398238996929238612315350783385370507310} a^{11} - \frac{79597226866199580611277242920960434520339234305744717483810963401595619333}{473915931424599014956265199057162053079647799385847722463070156677074101462} a^{10} - \frac{1016973240485254913715894871991084442362270114640419134378519053005373758993}{2369579657122995074781325995285810265398238996929238612315350783385370507310} a^{9} - \frac{30930217789689487844972785042422839542957423530247738311248174184535374524}{107708166232863412490060272512991375699919954405874482377970490153880477605} a^{8} + \frac{2078891054558497694937045735801293336023706671894820212878202532122549267}{11967574025873712498895585834776819522213328267319386930885610017097830845} a^{7} - \frac{43143604101249917219440137746345209561497645950640733995724611217851877757}{107708166232863412490060272512991375699919954405874482377970490153880477605} a^{6} - \frac{88038476692444578173113052071300682497114822812760627797383699001369435653}{215416332465726824980120545025982751399839908811748964755940980307760955210} a^{5} - \frac{24001811113788394834414741744542295877729930547001653372188156023446233009}{107708166232863412490060272512991375699919954405874482377970490153880477605} a^{4} + \frac{49638480526286376359344307241540270843775059763826204313245878515586333997}{107708166232863412490060272512991375699919954405874482377970490153880477605} a^{3} - \frac{3471009207439100365952949464610508914786540840634933753989496379281967441}{7180544415524227499337351500866091713327996960391632158531366010258698507} a^{2} + \frac{530963354282426034218514135382250123321931598198687093365412838714800829}{7978382683915808332597057223184546348142218844879591287257073344731887230} a + \frac{3464607270143023332543575015540487618590426681160646428221668604935342533}{7978382683915808332597057223184546348142218844879591287257073344731887230}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 40528597781300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T25:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12100
The 49 conjugacy class representatives for t22n25
Character table for t22n25 is not computed

Intermediate fields

\(\Q(\sqrt{-119}) \)

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

Sibling fields

Degree 44 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.11.10.1$x^{11} - 2$$11$$1$$10$$F_{11}$$[\ ]_{11}^{10}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.10.5.2$x^{10} - 83521 x^{2} + 8519142$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
17.10.5.2$x^{10} - 83521 x^{2} + 8519142$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$