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Properties

Label 20.0.11508250083...6304.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 41^{18}$
Root discriminant $79.99$
Ramified primes $2, 41$
Class number $31310$ (GRH)
Class group $[31310]$ (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![3286971, 734442, 4014037, -264554, 2036322, -477652, 766773, -154430, 179119, -40518, 20385, 906, 4287, 342, -203, -248, 174, 26, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 15*x^18 + 26*x^17 + 174*x^16 - 248*x^15 - 203*x^14 + 342*x^13 + 4287*x^12 + 906*x^11 + 20385*x^10 - 40518*x^9 + 179119*x^8 - 154430*x^7 + 766773*x^6 - 477652*x^5 + 2036322*x^4 - 264554*x^3 + 4014037*x^2 + 734442*x + 3286971)
 
gp: K = bnfinit(x^20 - 2*x^19 - 15*x^18 + 26*x^17 + 174*x^16 - 248*x^15 - 203*x^14 + 342*x^13 + 4287*x^12 + 906*x^11 + 20385*x^10 - 40518*x^9 + 179119*x^8 - 154430*x^7 + 766773*x^6 - 477652*x^5 + 2036322*x^4 - 264554*x^3 + 4014037*x^2 + 734442*x + 3286971, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 15 x^{18} + 26 x^{17} + 174 x^{16} - 248 x^{15} - 203 x^{14} + 342 x^{13} + 4287 x^{12} + 906 x^{11} + 20385 x^{10} - 40518 x^{9} + 179119 x^{8} - 154430 x^{7} + 766773 x^{6} - 477652 x^{5} + 2036322 x^{4} - 264554 x^{3} + 4014037 x^{2} + 734442 x + 3286971 \)

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:  \(115082500831447662912765425603842146304=2^{30}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.99$
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:  $2, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(328=2^{3}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{328}(1,·)$, $\chi_{328}(195,·)$, $\chi_{328}(201,·)$, $\chi_{328}(139,·)$, $\chi_{328}(305,·)$, $\chi_{328}(291,·)$, $\chi_{328}(81,·)$, $\chi_{328}(83,·)$, $\chi_{328}(25,·)$, $\chi_{328}(283,·)$, $\chi_{328}(163,·)$, $\chi_{328}(209,·)$, $\chi_{328}(297,·)$, $\chi_{328}(107,·)$, $\chi_{328}(113,·)$, $\chi_{328}(51,·)$, $\chi_{328}(59,·)$, $\chi_{328}(105,·)$, $\chi_{328}(57,·)$, $\chi_{328}(187,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{747} a^{17} - \frac{13}{249} a^{16} + \frac{107}{747} a^{15} - \frac{25}{747} a^{14} - \frac{12}{83} a^{13} + \frac{1}{747} a^{12} + \frac{55}{747} a^{11} - \frac{6}{83} a^{10} - \frac{35}{249} a^{9} + \frac{59}{747} a^{8} - \frac{226}{747} a^{7} + \frac{40}{83} a^{6} + \frac{295}{747} a^{5} - \frac{179}{747} a^{4} - \frac{73}{249} a^{3} + \frac{155}{747} a^{2} + \frac{16}{249} a - \frac{14}{83}$, $\frac{1}{747} a^{18} - \frac{1}{249} a^{16} + \frac{9}{83} a^{15} - \frac{29}{249} a^{14} + \frac{35}{249} a^{13} - \frac{8}{83} a^{12} + \frac{11}{83} a^{11} - \frac{53}{747} a^{10} + \frac{38}{249} a^{9} + \frac{4}{249} a^{7} - \frac{8}{249} a^{6} + \frac{68}{249} a^{5} - \frac{76}{249} a^{4} + \frac{82}{249} a^{3} + \frac{283}{747} a^{2} - \frac{82}{249} a + \frac{35}{83}$, $\frac{1}{1199733975647164170577154064678816134748452807817} a^{19} + \frac{378553869817960077299728937686714280540339647}{1199733975647164170577154064678816134748452807817} a^{18} - \frac{572850606286930052267048493513954422844897946}{1199733975647164170577154064678816134748452807817} a^{17} + \frac{1550499463027882077605044406155973733669031291}{1199733975647164170577154064678816134748452807817} a^{16} + \frac{11195980689836168691188544046910041115747875811}{1199733975647164170577154064678816134748452807817} a^{15} - \frac{68637402965154172157319347969996120489495587421}{1199733975647164170577154064678816134748452807817} a^{14} - \frac{64377956369540716943376312907841709382638871690}{1199733975647164170577154064678816134748452807817} a^{13} + \frac{178286494084504073180193451331087765266864882900}{1199733975647164170577154064678816134748452807817} a^{12} + \frac{32505628008376100703940310961730952870418055462}{399911325215721390192384688226272044916150935939} a^{11} - \frac{66736182097804846627259203168310220847081710535}{1199733975647164170577154064678816134748452807817} a^{10} - \frac{88757449145155391768577707755448229730821008279}{1199733975647164170577154064678816134748452807817} a^{9} - \frac{112710125555908246681056458705178129462026435881}{1199733975647164170577154064678816134748452807817} a^{8} + \frac{385310667373788928845694850198111370909355475572}{1199733975647164170577154064678816134748452807817} a^{7} - \frac{258635245736892414950805696240606323693474184967}{1199733975647164170577154064678816134748452807817} a^{6} + \frac{351941181970475074985404174578559134913406309158}{1199733975647164170577154064678816134748452807817} a^{5} - \frac{304653884249444757074697927864518915044949089861}{1199733975647164170577154064678816134748452807817} a^{4} - \frac{542281764313471658045406201397191906531699473722}{1199733975647164170577154064678816134748452807817} a^{3} + \frac{7962369923438704577375749091687418609186897659}{133303775071907130064128229408757348305383645313} a^{2} - \frac{117414094897390475200895864682688931548487098610}{399911325215721390192384688226272044916150935939} a - \frac{791562115058039703839885999027076115478920712}{1826079110574070274851071635736402031580597881}$

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

Class group and class number

$C_{31310}$, which has order $31310$ (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:  \( 5104264.63655 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-82}) \), \(\Q(\sqrt{-2}, \sqrt{41})\), 5.5.2825761.1, 10.10.327381934393961.1, 10.0.261650029907836928.1, 10.0.10727651226221314048.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$