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Properties

Label 20.0.91753302521...2013.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}\cdot 31^{10}$
Root discriminant $70.49$
Ramified primes $7, 11, 31$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![135033863407129, -34565255468696, 5543074970116, -1345126254829, 987102681652, -25786596024, 1197335012, 519847620, 803037260, 135093409, 21491823, -1325319, 1509397, -54957, -27121, -9300, -389, 114, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 114*x^17 - 389*x^16 - 9300*x^15 - 27121*x^14 - 54957*x^13 + 1509397*x^12 - 1325319*x^11 + 21491823*x^10 + 135093409*x^9 + 803037260*x^8 + 519847620*x^7 + 1197335012*x^6 - 25786596024*x^5 + 987102681652*x^4 - 1345126254829*x^3 + 5543074970116*x^2 - 34565255468696*x + 135033863407129)
 
gp: K = bnfinit(x^20 - 2*x^19 + 114*x^17 - 389*x^16 - 9300*x^15 - 27121*x^14 - 54957*x^13 + 1509397*x^12 - 1325319*x^11 + 21491823*x^10 + 135093409*x^9 + 803037260*x^8 + 519847620*x^7 + 1197335012*x^6 - 25786596024*x^5 + 987102681652*x^4 - 1345126254829*x^3 + 5543074970116*x^2 - 34565255468696*x + 135033863407129, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 114 x^{17} - 389 x^{16} - 9300 x^{15} - 27121 x^{14} - 54957 x^{13} + 1509397 x^{12} - 1325319 x^{11} + 21491823 x^{10} + 135093409 x^{9} + 803037260 x^{8} + 519847620 x^{7} + 1197335012 x^{6} - 25786596024 x^{5} + 987102681652 x^{4} - 1345126254829 x^{3} + 5543074970116 x^{2} - 34565255468696 x + 135033863407129 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9175330252180039242707571655982422013=7^{15}\cdot 11^{9}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.49$
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:  $7, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{16} + \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{7}{16} a^{11} + \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{7}{16} a^{7} + \frac{3}{8} a^{6} - \frac{7}{16} a^{5} + \frac{3}{8} a^{4} - \frac{7}{16} a^{3} - \frac{3}{8} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{3392} a^{18} - \frac{23}{848} a^{17} + \frac{3}{64} a^{16} + \frac{33}{424} a^{15} + \frac{177}{848} a^{14} + \frac{351}{848} a^{13} - \frac{1069}{3392} a^{12} - \frac{823}{3392} a^{11} + \frac{39}{212} a^{10} + \frac{51}{424} a^{9} + \frac{847}{3392} a^{8} - \frac{765}{3392} a^{7} - \frac{281}{3392} a^{6} - \frac{157}{3392} a^{5} - \frac{537}{3392} a^{4} - \frac{1017}{3392} a^{3} - \frac{177}{3392} a^{2} - \frac{353}{1696} a + \frac{1633}{3392}$, $\frac{1}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{19} - \frac{5099452358836476989711220234742545177132261263208092529014430572958729571319059528950316022704202731117377206947879111340739}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{18} + \frac{684886319521705526194851780357209233833213556224923197420979911195653888787612676939935110794213296445765537285699757719485363}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{17} - \frac{227590874106919509691322782484824902296127974262706829508548365503075241048656930095983238091258245954295979819329699641812993}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{16} - \frac{4266460222659965350082746540205564699565025955458664518292399446824911784962405934262141656903091814692180184416316718026240205}{13691366093274753971053751020455095620108987451868369304654483897483403001076828318551713538044491533349945578327772905726876352} a^{15} - \frac{248331294526846240012505425832503475053360362710429520975893910161906358996060560989438050174309443642686080573828333800648117}{1711420761659344246381718877556886952513623431483546163081810487185425375134603539818964192255561441668743197290971613215859544} a^{14} + \frac{10047029790610139797236379983884133033801261085583296723649009328741483139277573709425039970705248861399304883528007808727477871}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{13} + \frac{3978066086482988318257728623624001996452822868249507549972701187491667831768365973052036909439469071561130604492604543490047865}{13691366093274753971053751020455095620108987451868369304654483897483403001076828318551713538044491533349945578327772905726876352} a^{12} - \frac{5493260933165395256905754535016474881552035284106580357911415027750120796339857854274251268653813741433142141733336469033997247}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{11} - \frac{1299164306288006326908870804023292413984375858141143881852125689301155054052147988367887996737228184197109319835967485515531123}{6845683046637376985526875510227547810054493725934184652327241948741701500538414159275856769022245766674972789163886452863438176} a^{10} - \frac{22149611407426735493929194312510327626471981655687154961970798946911673661321827879298860636618480650101705514999108672417004665}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{9} - \frac{11724936906825529011182808841735301234453657180993818617431296587741665155266818225900531594967658503342059429748280881472990099}{27382732186549507942107502040910191240217974903736738609308967794966806002153656637103427076088983066699891156655545811453752704} a^{8} + \frac{11677810521845583679901036438568499636169790534616193893902406373186249649811106931989232751480019049334444508309710938367445601}{27382732186549507942107502040910191240217974903736738609308967794966806002153656637103427076088983066699891156655545811453752704} a^{7} + \frac{3290978951451666970351165540730812855958134624827499324338472894331062449799087088617130536026037885290044745609037376549264553}{27382732186549507942107502040910191240217974903736738609308967794966806002153656637103427076088983066699891156655545811453752704} a^{6} - \frac{6799349156835050899544139413679725883201734477784416359453859401057899384239155651944426347252290839559359253131560606016647759}{27382732186549507942107502040910191240217974903736738609308967794966806002153656637103427076088983066699891156655545811453752704} a^{5} + \frac{4412573551632965574216616704934329916255928273989162431265003874733263286173283469882054223152871328619115112573312106979028507}{27382732186549507942107502040910191240217974903736738609308967794966806002153656637103427076088983066699891156655545811453752704} a^{4} - \frac{6794335458056354049647977559025193099445238386275598186551727882157604875176138885948210974800327029811581596623269466773445913}{27382732186549507942107502040910191240217974903736738609308967794966806002153656637103427076088983066699891156655545811453752704} a^{3} - \frac{4502827013995224620961293753021040944417201914087689756747606510675538181329267946851286134625803696736155696724481041498250411}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a^{2} - \frac{12134097886539204884650953930019384361448143106442522537097252674127857885047713126136585312234900150385881516936408499357165889}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408} a + \frac{12316465959312776387715240952917987832856288470867682143967609040792232652786650403682526663794026752960455450114024646208430537}{54765464373099015884215004081820382480435949807473477218617935589933612004307313274206854152177966133399782313311091622907505408}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.3625853.1, 10.0.246071287.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
31Data not computed