↑

Properties

Label 20.0.26166731193...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{32}\cdot 41^{18}$
Root discriminant $742.83$
Ramified primes $2, 5, 41$
Class number $2530302500$ (GRH)
Class group $[5, 5, 5, 20242420]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1676212563480625, 0, 2670265073911875, 0, 671078556444375, 0, 73299560375500, 0, 4299563251375, 0, 146793173751, 0, 2990652340, 0, 35935475, 0, 242720, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 242720*x^16 + 35935475*x^14 + 2990652340*x^12 + 146793173751*x^10 + 4299563251375*x^8 + 73299560375500*x^6 + 671078556444375*x^4 + 2670265073911875*x^2 + 1676212563480625)
 
gp: K = bnfinit(x^20 + 820*x^18 + 242720*x^16 + 35935475*x^14 + 2990652340*x^12 + 146793173751*x^10 + 4299563251375*x^8 + 73299560375500*x^6 + 671078556444375*x^4 + 2670265073911875*x^2 + 1676212563480625, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 242720 x^{16} + 35935475 x^{14} + 2990652340 x^{12} + 146793173751 x^{10} + 4299563251375 x^{8} + 73299560375500 x^{6} + 671078556444375 x^{4} + 2670265073911875 x^{2} + 1676212563480625 \)

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:  \(2616673119324506454012530994165039062500000000000000000000=2^{20}\cdot 5^{32}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $742.83$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(2051,·)$, $\chi_{4100}(901,·)$, $\chi_{4100}(2951,·)$, $\chi_{4100}(271,·)$, $\chi_{4100}(2321,·)$, $\chi_{4100}(1111,·)$, $\chi_{4100}(3161,·)$, $\chi_{4100}(1691,·)$, $\chi_{4100}(221,·)$, $\chi_{4100}(2271,·)$, $\chi_{4100}(3741,·)$, $\chi_{4100}(611,·)$, $\chi_{4100}(2661,·)$, $\chi_{4100}(1581,·)$, $\chi_{4100}(3631,·)$, $\chi_{4100}(1781,·)$, $\chi_{4100}(3831,·)$, $\chi_{4100}(441,·)$, $\chi_{4100}(2491,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10}$, $\frac{1}{41} a^{11}$, $\frac{1}{41} a^{12}$, $\frac{1}{205} a^{13} + \frac{1}{5} a^{3}$, $\frac{1}{12505} a^{14} - \frac{4}{2501} a^{12} + \frac{21}{2501} a^{10} + \frac{12}{61} a^{8} + \frac{4}{61} a^{6} - \frac{54}{305} a^{4} - \frac{12}{61} a^{2} + \frac{22}{61}$, $\frac{1}{62525} a^{15} - \frac{4}{12505} a^{13} - \frac{101}{12505} a^{11} - \frac{22}{61} a^{9} - \frac{57}{305} a^{7} - \frac{664}{1525} a^{5} - \frac{73}{305} a^{3} - \frac{20}{61} a$, $\frac{1}{62525} a^{16} + \frac{124}{12505} a^{12} - \frac{25}{2501} a^{10} - \frac{2}{5} a^{8} - \frac{264}{1525} a^{6} + \frac{16}{305} a^{4} - \frac{7}{61} a^{2} + \frac{27}{61}$, $\frac{1}{62525} a^{17} + \frac{2}{12505} a^{13} - \frac{25}{2501} a^{11} - \frac{2}{5} a^{9} - \frac{264}{1525} a^{7} + \frac{16}{305} a^{5} + \frac{148}{305} a^{3} + \frac{27}{61} a$, $\frac{1}{266250583889988621457375619365948201413788944608875} a^{18} + \frac{330939979341179695187773479438689719691502318}{53250116777997724291475123873189640282757788921775} a^{16} + \frac{602607396029342804229609720704367322467636849}{53250116777997724291475123873189640282757788921775} a^{14} - \frac{21310945617090517143775267536659738488933919591}{2130004671119908971659004954927585611310311556871} a^{12} - \frac{589381393777636100006727368134399542664142294482}{53250116777997724291475123873189640282757788921775} a^{10} + \frac{1560850244845723057667471037198972738106616241861}{6493916680243624913594527301608492717409486453875} a^{8} + \frac{67856053229346884385045391696999659739799760794}{1298783336048724982718905460321698543481897290775} a^{6} + \frac{584668017153410681837033233112404426940947450}{51951333441948999308756218412867941739275891631} a^{4} - \frac{10398043220815878298946610743606510922194774684}{51951333441948999308756218412867941739275891631} a^{2} + \frac{23163394790570328914238198402517068373790211640}{51951333441948999308756218412867941739275891631}$, $\frac{1}{436028729965030435781750128937696829371909844395004059125} a^{19} - \frac{16428328830559730012534070997275790484880294278882}{2126969414463563101374390872866813801814194362902458825} a^{17} - \frac{55060073121826009350435578780582865526244873397}{425393882892712620274878174573362760362838872580491765} a^{15} - \frac{48348227294631678021557906056145760727404799167382}{425393882892712620274878174573362760362838872580491765} a^{13} - \frac{10752097485882903360944295531604354285724229501873082}{2126969414463563101374390872866813801814194362902458825} a^{11} + \frac{1269102055414780185338960391959680254504696622939950236}{10634847072317815506871954364334069009070971814512294125} a^{9} - \frac{4705712945864220133929037439607329242073895177318056}{51877302791794221984741240801629604922297423485425825} a^{7} + \frac{18870091080319743404627911104234602998078575368716761}{51877302791794221984741240801629604922297423485425825} a^{5} + \frac{203669437518508300569974146355796232244081903776354}{10375460558358844396948248160325920984459484697085165} a^{3} + \frac{325760684666844949685582903319810583653011044319706}{2075092111671768879389649632065184196891896939417033} a$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}\times C_{20242420}$, which has order $2530302500$ (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:  \( -\frac{1810450654925525751996}{7218889238543289361514467727829325} a^{19} - \frac{573374630256589032018}{2886401134963330412440810766825} a^{17} - \frac{9742902771984749129680199}{176070469232763155158889456776325} a^{15} - \frac{263223215349915587129501818}{35214093846552631031777891355265} a^{13} - \frac{19135205855862626470481781104}{35214093846552631031777891355265} a^{11} - \frac{3857166561739586008893016206676}{176070469232763155158889456776325} a^{9} - \frac{2078643703182138221812423258543}{4294401688603979394119255043325} a^{7} - \frac{23439688309526065883823748113734}{4294401688603979394119255043325} a^{5} - \frac{4373456783202356479835160608165}{171776067544159175764770201733} a^{3} - \frac{2627129129154079863267916496015}{171776067544159175764770201733} a \) (order $4$)
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:  \( 408291769676.07916 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-41}) \), \(\Q(\sqrt{41}) \), \(\Q(i, \sqrt{41})\), 5.5.1103812890625.2, 10.0.1247644567050156250000000000.2, 10.0.51153427249056406250000000000.4, 10.10.49954518797906646728515625.3

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$
5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$
5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$
5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$
$41$41.10.9.3$x^{10} - 53136$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.3$x^{10} - 53136$$10$$1$$9$$C_{10}$$[\ ]_{10}$