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Properties

Label 20.0.14359290582...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{35}\cdot 11^{16}$
Root discriminant $322.00$
Ramified primes $2, 5, 11$
Class number $183427210$ (GRH)
Class group $[183427210]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16039254706751, -9509061581690, 3562312190225, 155862360940, -187558748575, 54801750652, -21092572535, -4276903400, 5074550610, 238909550, -414134351, -6968280, 20953775, 120230, -676800, -902, 13970, 0, -170, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 170*x^18 + 13970*x^16 - 902*x^15 - 676800*x^14 + 120230*x^13 + 20953775*x^12 - 6968280*x^11 - 414134351*x^10 + 238909550*x^9 + 5074550610*x^8 - 4276903400*x^7 - 21092572535*x^6 + 54801750652*x^5 - 187558748575*x^4 + 155862360940*x^3 + 3562312190225*x^2 - 9509061581690*x + 16039254706751)
 
gp: K = bnfinit(x^20 - 170*x^18 + 13970*x^16 - 902*x^15 - 676800*x^14 + 120230*x^13 + 20953775*x^12 - 6968280*x^11 - 414134351*x^10 + 238909550*x^9 + 5074550610*x^8 - 4276903400*x^7 - 21092572535*x^6 + 54801750652*x^5 - 187558748575*x^4 + 155862360940*x^3 + 3562312190225*x^2 - 9509061581690*x + 16039254706751, 1)
 

Normalized defining polynomial

\( x^{20} - 170 x^{18} + 13970 x^{16} - 902 x^{15} - 676800 x^{14} + 120230 x^{13} + 20953775 x^{12} - 6968280 x^{11} - 414134351 x^{10} + 238909550 x^{9} + 5074550610 x^{8} - 4276903400 x^{7} - 21092572535 x^{6} + 54801750652 x^{5} - 187558748575 x^{4} + 155862360940 x^{3} + 3562312190225 x^{2} - 9509061581690 x + 16039254706751 \)

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:  \(143592905823663003125000000000000000000000000000000=2^{30}\cdot 5^{35}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $322.00$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(1797,·)$, $\chi_{2200}(521,·)$, $\chi_{2200}(653,·)$, $\chi_{2200}(333,·)$, $\chi_{2200}(1929,·)$, $\chi_{2200}(1169,·)$, $\chi_{2200}(1237,·)$, $\chi_{2200}(889,·)$, $\chi_{2200}(1373,·)$, $\chi_{2200}(1413,·)$, $\chi_{2200}(1849,·)$, $\chi_{2200}(1893,·)$, $\chi_{2200}(1809,·)$, $\chi_{2200}(361,·)$, $\chi_{2200}(2157,·)$, $\chi_{2200}(2077,·)$, $\chi_{2200}(841,·)$, $\chi_{2200}(1081,·)$, $\chi_{2200}(1917,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{13} + \frac{1}{25} a^{12} + \frac{2}{25} a^{11} - \frac{2}{25} a^{10} - \frac{2}{5} a^{9} + \frac{4}{25} a^{8} - \frac{6}{25} a^{7} - \frac{2}{25} a^{6} - \frac{3}{25} a^{5} + \frac{1}{5} a^{4} + \frac{9}{25} a^{3} - \frac{11}{25} a^{2} - \frac{2}{25} a - \frac{8}{25}$, $\frac{1}{25} a^{14} + \frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{2}{25} a^{10} - \frac{11}{25} a^{9} - \frac{2}{5} a^{8} + \frac{4}{25} a^{7} - \frac{6}{25} a^{6} - \frac{2}{25} a^{5} + \frac{4}{25} a^{4} + \frac{1}{5} a^{3} + \frac{9}{25} a^{2} - \frac{11}{25} a - \frac{2}{25}$, $\frac{1}{31775} a^{15} + \frac{118}{31775} a^{14} - \frac{586}{31775} a^{13} - \frac{3173}{31775} a^{12} - \frac{754}{31775} a^{11} + \frac{74}{775} a^{10} - \frac{12138}{31775} a^{9} - \frac{6649}{31775} a^{8} - \frac{12832}{31775} a^{7} - \frac{5236}{31775} a^{6} - \frac{8281}{31775} a^{5} - \frac{10083}{31775} a^{4} + \frac{9091}{31775} a^{3} + \frac{3088}{31775} a^{2} + \frac{12174}{31775} a$, $\frac{1}{48583975} a^{16} + \frac{1}{107725} a^{15} - \frac{284126}{48583975} a^{14} + \frac{24608}{4416725} a^{13} - \frac{2970218}{48583975} a^{12} + \frac{317616}{4416725} a^{11} - \frac{317247}{4416725} a^{10} - \frac{1667479}{4416725} a^{9} - \frac{14408213}{48583975} a^{8} - \frac{1320232}{4416725} a^{7} + \frac{21318052}{48583975} a^{6} + \frac{78610}{176669} a^{5} + \frac{158824}{349525} a^{4} + \frac{1321292}{4416725} a^{3} + \frac{1771303}{48583975} a^{2} + \frac{1543753}{4416725} a + \frac{12641}{38225}$, $\frac{1}{48583975} a^{17} + \frac{224}{48583975} a^{15} - \frac{54212}{4416725} a^{14} - \frac{823843}{48583975} a^{13} - \frac{47194}{4416725} a^{12} - \frac{149508}{4416725} a^{11} - \frac{368804}{4416725} a^{10} - \frac{2170768}{48583975} a^{9} - \frac{620707}{4416725} a^{8} - \frac{8486063}{48583975} a^{7} + \frac{643568}{4416725} a^{6} + \frac{160411}{48583975} a^{5} - \frac{95813}{4416725} a^{4} + \frac{7020778}{48583975} a^{3} - \frac{405537}{4416725} a^{2} + \frac{13318108}{48583975} a + \frac{2}{139}$, $\frac{1}{11138313524525} a^{18} - \frac{3563}{445532540981} a^{17} + \frac{40483}{11138313524525} a^{16} - \frac{599317}{271666183525} a^{15} + \frac{19809067557}{2227662704905} a^{14} + \frac{193764044238}{11138313524525} a^{13} + \frac{6167033043}{80131751975} a^{12} - \frac{92308535988}{1012573956775} a^{11} + \frac{812022513934}{11138313524525} a^{10} - \frac{1010127436326}{2227662704905} a^{9} - \frac{1086157369336}{2227662704905} a^{8} + \frac{3825070228663}{11138313524525} a^{7} - \frac{501615432273}{11138313524525} a^{6} - \frac{168288518290}{445532540981} a^{5} + \frac{52174554442}{445532540981} a^{4} + \frac{4736946563906}{11138313524525} a^{3} - \frac{3514426255147}{11138313524525} a^{2} - \frac{633127705623}{2227662704905} a + \frac{584560733}{8763425275}$, $\frac{1}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{19} + \frac{249178828873836114381017090877609281074265551386826122333637578698973}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{18} + \frac{30541919236390609211274464178639666831190655072957785307977843635497846137}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{17} - \frac{37811944669605077655863423532771917864328866061920564440786361213570668788}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{16} - \frac{14602558211253561409004461770496404745006781738871154565330329778259137770583}{1383623856869386360693362337805117588854151595625959338302643683533534352532290255} a^{15} - \frac{25966668607550001087425852119745889452566436083925797635173867497454448854774884}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{14} - \frac{115340379583284610194433234750703000026863659259295389962555105171151996798749972}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{13} + \frac{272014946987450523691533366554325251791381377760001944552450031446180408240872522}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{12} - \frac{330373086508488797125625215803650448804482746351530027026495996286859119976207918}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{11} + \frac{13507954729249653947062331212715093266166688232152332058815938515906353896879227}{1383623856869386360693362337805117588854151595625959338302643683533534352532290255} a^{10} - \frac{400807280732183931924944931243152769102141358577576733683840599029382379797527418}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{9} + \frac{2807240710303863685920250387185492547164442612569850101415356576291149334307700001}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{8} + \frac{1002067632761567582047916556649018572923028484515032074073947177168701632467037504}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{7} - \frac{2529846367092847622424569391292962912833159696459139462429294613745858773182164906}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{6} - \frac{520791034519970113961750068237502885840692243652935136801718903245786569032148486}{1383623856869386360693362337805117588854151595625959338302643683533534352532290255} a^{5} - \frac{37772470356730402685352180968936130653674951771304322908094755679967849203048092}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{4} + \frac{159500892463903425959044761846179847910373989329983586477149865171078571639802409}{628919934940630163951528335365962540388250725284526971955747128878879251151041025} a^{3} + \frac{73233178390183122186026065319712571193693017271116944243677630536135864128823066}{6918119284346931803466811689025587944270757978129796691513218417667671762661451275} a^{2} - \frac{43800506091029772220780996603337220554525249310185913767080832736586543700884724}{223165138204739735595703602871793159492605096068703119081071561860247476214885525} a - \frac{16724346600272774648740714462794410515018369678507585222263024834709085436019}{35116465492484616144091833654098058142030699617419845646116689513807628043255}$

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

Class group and class number

$C_{183427210}$, which has order $183427210$ (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:  $9$
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:  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:  \( 860169583.1577827 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.8000.2, 5.5.5719140625.1, 10.10.163542847442626953125.2

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 R $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
5Data not computed
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$