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Properties

Label 20.0.66362369058...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $309.81$
Ramified primes $2, 3, 5, 41$
Class number $211132416$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 44, 18744]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![369595501344, -84572527680, 86704169904, -4121177184, 1964639016, -575898768, 1241919960, -706860960, -41725982, 111370892, 9489317, -14266122, -3887, 922480, -10862, -43484, 2452, 996, -79, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 79*x^18 + 996*x^17 + 2452*x^16 - 43484*x^15 - 10862*x^14 + 922480*x^13 - 3887*x^12 - 14266122*x^11 + 9489317*x^10 + 111370892*x^9 - 41725982*x^8 - 706860960*x^7 + 1241919960*x^6 - 575898768*x^5 + 1964639016*x^4 - 4121177184*x^3 + 86704169904*x^2 - 84572527680*x + 369595501344)
 
gp: K = bnfinit(x^20 - 10*x^19 - 79*x^18 + 996*x^17 + 2452*x^16 - 43484*x^15 - 10862*x^14 + 922480*x^13 - 3887*x^12 - 14266122*x^11 + 9489317*x^10 + 111370892*x^9 - 41725982*x^8 - 706860960*x^7 + 1241919960*x^6 - 575898768*x^5 + 1964639016*x^4 - 4121177184*x^3 + 86704169904*x^2 - 84572527680*x + 369595501344, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 79 x^{18} + 996 x^{17} + 2452 x^{16} - 43484 x^{15} - 10862 x^{14} + 922480 x^{13} - 3887 x^{12} - 14266122 x^{11} + 9489317 x^{10} + 111370892 x^{9} - 41725982 x^{8} - 706860960 x^{7} + 1241919960 x^{6} - 575898768 x^{5} + 1964639016 x^{4} - 4121177184 x^{3} + 86704169904 x^{2} - 84572527680 x + 369595501344 \)

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:  \(66362369058556182102889507973449950167040000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $309.81$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(901,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(1289,·)$, $\chi_{4920}(269,·)$, $\chi_{4920}(1229,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(4889,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(3809,·)$, $\chi_{4920}(1501,·)$, $\chi_{4920}(2341,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(1829,·)$, $\chi_{4920}(4649,·)$, $\chi_{4920}(2669,·)$, $\chi_{4920}(1261,·)$, $\chi_{4920}(1589,·)$, $\chi_{4920}(329,·)$, $\chi_{4920}(4861,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{4} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{36} a^{9} - \frac{1}{12} a^{5} - \frac{5}{18} a^{3} + \frac{1}{3} a$, $\frac{1}{36} a^{10} - \frac{1}{12} a^{6} + \frac{2}{9} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{36} a^{11} - \frac{1}{12} a^{7} + \frac{2}{9} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{216} a^{12} - \frac{1}{216} a^{10} - \frac{1}{72} a^{8} + \frac{17}{216} a^{6} + \frac{23}{108} a^{4} - \frac{5}{18} a^{2} - \frac{1}{3}$, $\frac{1}{216} a^{13} - \frac{1}{216} a^{11} - \frac{1}{72} a^{9} + \frac{17}{216} a^{7} + \frac{23}{108} a^{5} - \frac{5}{18} a^{3} - \frac{1}{3} a$, $\frac{1}{216} a^{14} + \frac{1}{108} a^{10} - \frac{1}{54} a^{8} + \frac{1}{24} a^{6} + \frac{13}{54} a^{4} + \frac{7}{18} a^{2} - \frac{1}{3}$, $\frac{1}{216} a^{15} + \frac{1}{108} a^{11} + \frac{1}{108} a^{9} + \frac{1}{24} a^{7} + \frac{17}{108} a^{5} + \frac{1}{9} a^{3}$, $\frac{1}{283824} a^{16} - \frac{1}{35478} a^{15} + \frac{13}{15768} a^{14} - \frac{23}{35478} a^{13} - \frac{305}{141912} a^{12} - \frac{61}{7884} a^{11} + \frac{176}{17739} a^{10} - \frac{331}{70956} a^{9} + \frac{457}{31536} a^{8} - \frac{1835}{70956} a^{7} - \frac{8569}{141912} a^{6} + \frac{431}{7884} a^{5} - \frac{181}{1971} a^{4} + \frac{8}{657} a^{3} - \frac{1}{438} a^{2} - \frac{151}{657} a + \frac{29}{657}$, $\frac{1}{283824} a^{17} + \frac{85}{141912} a^{15} + \frac{187}{141912} a^{14} + \frac{91}{47304} a^{13} - \frac{253}{141912} a^{12} - \frac{403}{70956} a^{11} - \frac{209}{15768} a^{10} + \frac{1405}{283824} a^{9} + \frac{5555}{141912} a^{8} + \frac{2687}{47304} a^{7} - \frac{5431}{70956} a^{6} - \frac{463}{3942} a^{5} - \frac{25}{2628} a^{4} + \frac{106}{219} a^{3} + \frac{185}{1314} a^{2} + \frac{15}{73} a + \frac{13}{657}$, $\frac{1}{45197424217200125454856652553157008} a^{18} - \frac{1}{5021936024133347272761850283684112} a^{17} + \frac{10798204229970941447325315347}{11299356054300031363714163138289252} a^{16} - \frac{86385633839767531578602522725}{11299356054300031363714163138289252} a^{15} + \frac{6672763800246539314482677291449}{7532904036200020909142775425526168} a^{14} - \frac{4060109431417590893709990072749}{2824839013575007840928540784572313} a^{13} + \frac{16036241032625870077445075175719}{11299356054300031363714163138289252} a^{12} + \frac{8220570214685191357142416144097}{836989337355557878793641713947352} a^{11} - \frac{170658311150759019220681323966875}{45197424217200125454856652553157008} a^{10} - \frac{119527256110091763066002290033199}{45197424217200125454856652553157008} a^{9} + \frac{65554593555025449078955578566305}{3766452018100010454571387712763084} a^{8} - \frac{950730046464341143845300248369237}{22598712108600062727428326276578504} a^{7} + \frac{241940703866708200258364542917235}{3766452018100010454571387712763084} a^{6} - \frac{26106416642354966077502897727470}{104623667169444734849205214243419} a^{5} - \frac{84787125530376770784414098190805}{418494668677778939396820856973676} a^{4} - \frac{39670701683560073742519925932995}{104623667169444734849205214243419} a^{3} - \frac{233183874724099725446581292769}{7749901271810721099941126980994} a^{2} + \frac{693991590952935156538319838946}{1433200920129379929441167318403} a - \frac{6001437985875143233417787010964}{34874555723148244949735071414473}$, $\frac{1}{9108547818369209181469989666575549549777232} a^{19} + \frac{100764005}{9108547818369209181469989666575549549777232} a^{18} + \frac{6973324453562927545264791422518396243}{4554273909184604590734994833287774774888616} a^{17} + \frac{1181227033863080113824137065365888175}{759045651530767431789165805547962462481436} a^{16} + \frac{6245643045351261555745434864227013993365}{4554273909184604590734994833287774774888616} a^{15} + \frac{2391749881043884502215801734983989726367}{1138568477296151147683748708321943693722154} a^{14} - \frac{9273816616284654820184073387925632075613}{4554273909184604590734994833287774774888616} a^{13} - \frac{550410369022494112846140473490681163289}{569284238648075573841874354160971846861077} a^{12} + \frac{20460809537136655410561596376153652152499}{9108547818369209181469989666575549549777232} a^{11} + \frac{26036157366978702524680733592243799936775}{3036182606123069727156663222191849849925744} a^{10} - \frac{26276213877433686393435032360391951384751}{2277136954592302295367497416643887387444308} a^{9} + \frac{42943053823708039987248326633819145853913}{2277136954592302295367497416643887387444308} a^{8} + \frac{170826073454785252273112934063512651516537}{4554273909184604590734994833287774774888616} a^{7} + \frac{30823066080238853081175241868039848507357}{1518091303061534863578331611095924924962872} a^{6} - \frac{3005131415814307629191370515132541127475}{21084601431410206438587939042998957291151} a^{5} - \frac{4886856532739483817758593015208627394811}{21084601431410206438587939042998957291151} a^{4} + \frac{3423545737398869906734588702101217510124}{21084601431410206438587939042998957291151} a^{3} - \frac{15042491558225720403018926538142615385011}{42169202862820412877175878085997914582302} a^{2} - \frac{3496677322923201094653248699735152363087}{21084601431410206438587939042998957291151} a - \frac{5187483359331583321907866326723846176}{64478903459970050270911128571862254713}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{44}\times C_{18744}$, which has order $211132416$ (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:  \( 6411717617.202166 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1230}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{-15}, \sqrt{82})\), 5.5.2825761.1, 10.0.6063552595863759375.1, 10.0.8146310149911810355200000.1, 10.10.10727651226221314048.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
5Data not computed
41Data not computed