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Properties

Label 22.6.80119853675...0000.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{48}\cdot 3^{16}\cdot 5^{27}\cdot 31^{6}$
Root discriminant $185.50$
Ramified primes $2, 3, 5, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T48

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![549451881, -3243825927, 7469116746, -10215692655, 12781094360, -15698920969, 11379240893, -7396721954, 3962459515, -1085639155, 285023156, -22310537, -18440204, 4298285, -1607845, -22154, 11653, -14689, -520, 265, -14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^21 - 14*x^20 + 265*x^19 - 520*x^18 - 14689*x^17 + 11653*x^16 - 22154*x^15 - 1607845*x^14 + 4298285*x^13 - 18440204*x^12 - 22310537*x^11 + 285023156*x^10 - 1085639155*x^9 + 3962459515*x^8 - 7396721954*x^7 + 11379240893*x^6 - 15698920969*x^5 + 12781094360*x^4 - 10215692655*x^3 + 7469116746*x^2 - 3243825927*x + 549451881)
 
gp: K = bnfinit(x^22 - 7*x^21 - 14*x^20 + 265*x^19 - 520*x^18 - 14689*x^17 + 11653*x^16 - 22154*x^15 - 1607845*x^14 + 4298285*x^13 - 18440204*x^12 - 22310537*x^11 + 285023156*x^10 - 1085639155*x^9 + 3962459515*x^8 - 7396721954*x^7 + 11379240893*x^6 - 15698920969*x^5 + 12781094360*x^4 - 10215692655*x^3 + 7469116746*x^2 - 3243825927*x + 549451881, 1)
 

Normalized defining polynomial

\( x^{22} - 7 x^{21} - 14 x^{20} + 265 x^{19} - 520 x^{18} - 14689 x^{17} + 11653 x^{16} - 22154 x^{15} - 1607845 x^{14} + 4298285 x^{13} - 18440204 x^{12} - 22310537 x^{11} + 285023156 x^{10} - 1085639155 x^{9} + 3962459515 x^{8} - 7396721954 x^{7} + 11379240893 x^{6} - 15698920969 x^{5} + 12781094360 x^{4} - 10215692655 x^{3} + 7469116746 x^{2} - 3243825927 x + 549451881 \)

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:  \(80119853675928619057152000000000000000000000000000=2^{48}\cdot 3^{16}\cdot 5^{27}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $185.50$
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:  $2, 3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{40} a^{10} - \frac{1}{8} a^{6} + \frac{1}{5} a^{5} - \frac{3}{8} a^{2} + \frac{3}{20}$, $\frac{1}{40} a^{11} - \frac{1}{8} a^{7} - \frac{1}{20} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{3} - \frac{1}{10} a + \frac{1}{4}$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{10} + \frac{1}{10} a^{7} - \frac{1}{16} a^{6} + \frac{3}{20} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{11}{80} a^{2} + \frac{1}{4} a - \frac{1}{80}$, $\frac{1}{400} a^{13} + \frac{1}{200} a^{12} + \frac{3}{400} a^{11} + \frac{1}{100} a^{10} + \frac{1}{20} a^{9} - \frac{11}{200} a^{8} - \frac{49}{400} a^{7} - \frac{9}{100} a^{6} + \frac{23}{100} a^{5} - \frac{7}{40} a^{4} - \frac{149}{400} a^{3} + \frac{33}{100} a^{2} - \frac{177}{400} a - \frac{11}{25}$, $\frac{1}{800} a^{14} - \frac{1}{800} a^{13} + \frac{1}{400} a^{12} + \frac{1}{160} a^{11} + \frac{3}{800} a^{10} + \frac{9}{400} a^{9} + \frac{17}{800} a^{8} + \frac{1}{800} a^{7} + \frac{11}{160} a^{6} - \frac{43}{400} a^{5} + \frac{61}{800} a^{4} + \frac{29}{800} a^{3} - \frac{9}{400} a^{2} + \frac{3}{160} a - \frac{377}{800}$, $\frac{1}{800} a^{15} - \frac{1}{800} a^{13} + \frac{3}{800} a^{12} + \frac{1}{400} a^{11} - \frac{7}{800} a^{10} - \frac{1}{160} a^{9} - \frac{19}{400} a^{8} - \frac{23}{400} a^{7} - \frac{59}{800} a^{6} - \frac{169}{800} a^{5} + \frac{13}{80} a^{4} - \frac{291}{800} a^{3} + \frac{33}{800} a^{2} + \frac{49}{100} a - \frac{9}{160}$, $\frac{1}{1600} a^{16} - \frac{1}{160} a^{12} - \frac{1}{200} a^{11} + \frac{1}{40} a^{9} + \frac{3}{320} a^{8} - \frac{1}{40} a^{7} - \frac{3}{25} a^{6} + \frac{1}{40} a^{5} - \frac{9}{160} a^{4} + \frac{19}{40} a^{3} + \frac{3}{8} a^{2} - \frac{47}{200} a - \frac{3}{320}$, $\frac{1}{1600} a^{17} - \frac{1}{800} a^{13} + \frac{1}{200} a^{12} - \frac{1}{100} a^{11} - \frac{1}{200} a^{10} - \frac{1}{64} a^{9} - \frac{1}{100} a^{8} + \frac{1}{100} a^{7} - \frac{21}{200} a^{6} - \frac{97}{800} a^{5} + \frac{1}{200} a^{3} - \frac{3}{40} a^{2} - \frac{671}{1600} a - \frac{61}{200}$, $\frac{1}{99200} a^{18} + \frac{3}{99200} a^{17} + \frac{1}{3968} a^{16} + \frac{11}{49600} a^{14} - \frac{59}{49600} a^{13} - \frac{13}{49600} a^{12} - \frac{23}{12400} a^{11} + \frac{159}{99200} a^{10} + \frac{5421}{99200} a^{9} + \frac{73}{3200} a^{8} + \frac{1211}{12400} a^{7} - \frac{1501}{49600} a^{6} + \frac{41}{49600} a^{5} - \frac{2529}{49600} a^{4} + \frac{1619}{6200} a^{3} - \frac{1231}{3968} a^{2} - \frac{15109}{99200} a + \frac{38729}{99200}$, $\frac{1}{99200} a^{19} + \frac{1}{6200} a^{17} - \frac{13}{99200} a^{16} + \frac{11}{49600} a^{15} - \frac{3}{4960} a^{14} - \frac{11}{24800} a^{13} + \frac{133}{49600} a^{12} + \frac{91}{99200} a^{11} + \frac{151}{24800} a^{10} - \frac{231}{12400} a^{9} - \frac{1007}{99200} a^{8} - \frac{247}{1984} a^{7} - \frac{1417}{24800} a^{6} + \frac{162}{775} a^{5} + \frac{11611}{49600} a^{4} + \frac{3981}{99200} a^{3} + \frac{1943}{6200} a^{2} - \frac{51}{992} a - \frac{24737}{99200}$, $\frac{1}{1190400} a^{20} + \frac{1}{595200} a^{19} - \frac{1}{238080} a^{18} - \frac{23}{119040} a^{17} + \frac{43}{238080} a^{16} - \frac{97}{297600} a^{15} + \frac{49}{119040} a^{14} + \frac{13}{297600} a^{13} - \frac{7387}{1190400} a^{12} + \frac{37}{19200} a^{11} + \frac{733}{1190400} a^{10} - \frac{1123}{19200} a^{9} - \frac{15139}{1190400} a^{8} - \frac{24199}{297600} a^{7} + \frac{881}{595200} a^{6} + \frac{62491}{297600} a^{5} - \frac{1831}{38400} a^{4} + \frac{61501}{595200} a^{3} + \frac{42803}{1190400} a^{2} + \frac{65979}{198400} a - \frac{1289}{79360}$, $\frac{1}{3896992108214149065966120013647309952991698766232014187514446283957798567739942400} a^{21} - \frac{25127477985140023470119563895454825209224900006062362011834882803440180089}{60890501690846079155720625213239218015495293222375221679913223186840602620936600} a^{20} - \frac{7396236002081974474043488685404914214733284213411090006285696062135415065941}{3896992108214149065966120013647309952991698766232014187514446283957798567739942400} a^{19} - \frac{100560702808955198220545456256395033479792250274280728198905875818958334261}{24356200676338431662288250085295687206198117288950088671965289274736241048374640} a^{18} + \frac{697609575138263487455720735902241356129224695852919578044372982883718691530007}{3896992108214149065966120013647309952991698766232014187514446283957798567739942400} a^{17} + \frac{34134948293197528661032496329720450331162954163220811022032843973005614124527}{1948496054107074532983060006823654976495849383116007093757223141978899283869971200} a^{16} - \frac{175026289546580217448849680885750052355359926201354807837782256480535401215611}{1948496054107074532983060006823654976495849383116007093757223141978899283869971200} a^{15} - \frac{476710024626039532622158137989398737696874745930598405098060804063752392127}{19484960541070745329830600068236549764958493831160070937572231419788992838699712} a^{14} + \frac{76645616872984788177780244523388131379424009098322034974510437327018345119549}{155879684328565962638644800545892398119667950649280567500577851358311942709597696} a^{13} - \frac{5687997454477279064739262767757318607507951893627751644862043866477823722397037}{974248027053537266491530003411827488247924691558003546878611570989449641934985600} a^{12} + \frac{45037882543999314042017285019336584542828313958663184479970800463170431108626333}{3896992108214149065966120013647309952991698766232014187514446283957798567739942400} a^{11} + \frac{81072212736788538068258432203838990350295622588340000729429869493503339602731}{7856838927851106987834919382353447485870360415790351184504932024108464854314400} a^{10} + \frac{144003289080949936772495340368447082144251878167916803088193545532774470789786493}{3896992108214149065966120013647309952991698766232014187514446283957798567739942400} a^{9} - \frac{16344236597989800962318401514210374999040528631723400318664733274471700724138391}{389699210821414906596612001364730995299169876623201418751444628395779856773994240} a^{8} - \frac{1362769889487982554719071839630718004031397838282963787367844465621241186689023}{77939842164282981319322400272946199059833975324640283750288925679155971354798848} a^{7} - \frac{2194794607533984571017987835027622268745972926607072816964844565702497671237447}{243562006763384316622882500852956872061981172889500886719652892747362410483746400} a^{6} + \frac{782374691085963101271846544164053624453486286663997273098599055667248245226116903}{3896992108214149065966120013647309952991698766232014187514446283957798567739942400} a^{5} - \frac{42215773360711295379912205775303613661550530690284912872825418282803005036482373}{974248027053537266491530003411827488247924691558003546878611570989449641934985600} a^{4} - \frac{315641546599832033301492000945520553681320414936563229903111989707332382798245001}{779398421642829813193224002729461990598339753246402837502889256791559713547988480} a^{3} - \frac{25642853524784703825759636223812317444701040962408728335859416033477864246831577}{54124890391863181471751666856212638235995816197666863715478420610524980107499200} a^{2} - \frac{162124874951718500745157206754684014527340740160963442442965673000557440382474877}{1298997369404716355322040004549103317663899588744004729171482094652599522579980800} a - \frac{3414274242112249445346269594189593275136374328873781650841005636256642515787927}{6983856824756539544742150562091953320773653702924756608448828465874190981612800}$

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:  \( 129933926047000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 125452800
The 65 conjugacy class representatives for t22n48 are not computed
Character table for t22n48 is not computed

Intermediate fields

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

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

Sibling fields

Degree 24 sibling: data not computed
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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.13.6$x^{10} - 10 x^{5} + 15 x^{4} + 5$$10$$1$$13$$D_5\times C_5$$[3/2]_{2}^{5}$
5.10.13.6$x^{10} - 10 x^{5} + 15 x^{4} + 5$$10$$1$$13$$D_5\times C_5$$[3/2]_{2}^{5}$
31Data not computed