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Properties

Label 20.0.29133682277...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{16}$
Root discriminant $105.49$
Ramified primes $2, 3, 5, 11$
Class number $1026100$ (GRH)
Class group $[10, 102610]$ (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![654850801, -317690032, 580187432, -191903184, 246924546, -147284512, 111075844, -20356532, 15872079, -7761452, 5226170, -1738516, 886294, -182040, 75234, -10164, 3530, -304, 90, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 90*x^18 - 304*x^17 + 3530*x^16 - 10164*x^15 + 75234*x^14 - 182040*x^13 + 886294*x^12 - 1738516*x^11 + 5226170*x^10 - 7761452*x^9 + 15872079*x^8 - 20356532*x^7 + 111075844*x^6 - 147284512*x^5 + 246924546*x^4 - 191903184*x^3 + 580187432*x^2 - 317690032*x + 654850801)
 
gp: K = bnfinit(x^20 - 4*x^19 + 90*x^18 - 304*x^17 + 3530*x^16 - 10164*x^15 + 75234*x^14 - 182040*x^13 + 886294*x^12 - 1738516*x^11 + 5226170*x^10 - 7761452*x^9 + 15872079*x^8 - 20356532*x^7 + 111075844*x^6 - 147284512*x^5 + 246924546*x^4 - 191903184*x^3 + 580187432*x^2 - 317690032*x + 654850801, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 90 x^{18} - 304 x^{17} + 3530 x^{16} - 10164 x^{15} + 75234 x^{14} - 182040 x^{13} + 886294 x^{12} - 1738516 x^{11} + 5226170 x^{10} - 7761452 x^{9} + 15872079 x^{8} - 20356532 x^{7} + 111075844 x^{6} - 147284512 x^{5} + 246924546 x^{4} - 191903184 x^{3} + 580187432 x^{2} - 317690032 x + 654850801 \)

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:  \(29133682277961802980800184975360000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.49$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(1219,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(911,·)$, $\chi_{1320}(1109,·)$, $\chi_{1320}(71,·)$, $\chi_{1320}(859,·)$, $\chi_{1320}(551,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(619,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(499,·)$, $\chi_{1320}(311,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(379,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{115} a^{14} + \frac{11}{115} a^{13} - \frac{4}{115} a^{12} + \frac{8}{115} a^{11} + \frac{2}{115} a^{10} + \frac{2}{23} a^{9} - \frac{49}{115} a^{8} + \frac{19}{115} a^{7} + \frac{11}{115} a^{6} - \frac{18}{115} a^{5} - \frac{36}{115} a^{4} - \frac{4}{115} a^{3} + \frac{7}{23} a^{2} + \frac{4}{23} a - \frac{52}{115}$, $\frac{1}{115} a^{15} - \frac{2}{23} a^{13} + \frac{6}{115} a^{12} + \frac{6}{115} a^{11} + \frac{11}{115} a^{10} + \frac{5}{23} a^{9} - \frac{8}{23} a^{8} - \frac{37}{115} a^{7} + \frac{9}{23} a^{6} + \frac{24}{115} a^{5} + \frac{1}{115} a^{4} + \frac{56}{115} a^{3} - \frac{43}{115} a^{2} + \frac{27}{115} a + \frac{43}{115}$, $\frac{1}{115} a^{16} + \frac{1}{115} a^{13} - \frac{11}{115} a^{12} - \frac{1}{115} a^{11} - \frac{1}{115} a^{10} - \frac{32}{115} a^{9} - \frac{44}{115} a^{8} + \frac{1}{23} a^{7} - \frac{10}{23} a^{6} + \frac{51}{115} a^{5} - \frac{1}{23} a^{4} + \frac{9}{115} a^{3} + \frac{11}{23} a^{2} - \frac{56}{115} a - \frac{37}{115}$, $\frac{1}{115} a^{17} + \frac{1}{115} a^{13} + \frac{3}{115} a^{12} - \frac{9}{115} a^{11} - \frac{11}{115} a^{10} - \frac{31}{115} a^{9} + \frac{8}{115} a^{8} - \frac{2}{5} a^{7} - \frac{52}{115} a^{6} + \frac{36}{115} a^{5} - \frac{24}{115} a^{4} - \frac{56}{115} a^{3} + \frac{24}{115} a^{2} + \frac{12}{115} a + \frac{6}{115}$, $\frac{1}{128237567379445456460838535} a^{18} - \frac{96344726842495426799748}{128237567379445456460838535} a^{17} + \frac{417919810742276570861286}{128237567379445456460838535} a^{16} + \frac{374414561357235946233439}{128237567379445456460838535} a^{15} + \frac{344578626235328970814253}{128237567379445456460838535} a^{14} - \frac{12191285615551365395508856}{128237567379445456460838535} a^{13} - \frac{10947026465743746614233586}{128237567379445456460838535} a^{12} - \frac{8604316893172474506018396}{128237567379445456460838535} a^{11} - \frac{948852622922640515500791}{128237567379445456460838535} a^{10} + \frac{43275393628385426259045869}{128237567379445456460838535} a^{9} + \frac{41878323065720147393907572}{128237567379445456460838535} a^{8} + \frac{42395401881338320529173822}{128237567379445456460838535} a^{7} - \frac{9619080042629518550764563}{128237567379445456460838535} a^{6} - \frac{619905219827808500049753}{25647513475889091292167707} a^{5} + \frac{62700004504333821986833312}{128237567379445456460838535} a^{4} + \frac{37954471312190862578800008}{128237567379445456460838535} a^{3} - \frac{14231565694222702669502186}{128237567379445456460838535} a^{2} + \frac{41881137675889317604172184}{128237567379445456460838535} a - \frac{20853320196400593814660}{97519062645966126586189}$, $\frac{1}{8397378485568779601424058120496725082521164075741015} a^{19} + \frac{13558505463018166641360457}{8397378485568779601424058120496725082521164075741015} a^{18} + \frac{26570885466217608367874641391257592615636375847006}{8397378485568779601424058120496725082521164075741015} a^{17} - \frac{33365680450602388437274767901001710320649719362922}{8397378485568779601424058120496725082521164075741015} a^{16} + \frac{32322367780585552222672171484874574280047032811511}{8397378485568779601424058120496725082521164075741015} a^{15} - \frac{25034400724418963067546656208484321932806101676494}{8397378485568779601424058120496725082521164075741015} a^{14} - \frac{251680856327989117721802938206799428830018684877209}{8397378485568779601424058120496725082521164075741015} a^{13} - \frac{376058336442163025589109383141738641771587122047009}{8397378485568779601424058120496725082521164075741015} a^{12} + \frac{323318872283544124163915407233805286138250292179199}{8397378485568779601424058120496725082521164075741015} a^{11} - \frac{66473596717502693675890742714563488075338002670681}{1679475697113755920284811624099345016504232815148203} a^{10} + \frac{201703562007909075384609317388609625077193969537484}{8397378485568779601424058120496725082521164075741015} a^{9} - \frac{3849634397634089682937150060991039397974620873700746}{8397378485568779601424058120496725082521164075741015} a^{8} + \frac{878690437925624206089875835681604427940758361264027}{8397378485568779601424058120496725082521164075741015} a^{7} + \frac{338669837472256959135146445187443714435746657116614}{8397378485568779601424058120496725082521164075741015} a^{6} - \frac{3906614303612629000867096020176664947500657098741988}{8397378485568779601424058120496725082521164075741015} a^{5} - \frac{2432996896347061960090586160057307079742612033902143}{8397378485568779601424058120496725082521164075741015} a^{4} + \frac{2934666085604783996308709758358999368249161768474878}{8397378485568779601424058120496725082521164075741015} a^{3} + \frac{935409925540154134651761771174371714238279288110028}{8397378485568779601424058120496725082521164075741015} a^{2} + \frac{1865433515770296250888764838375378361804337170340284}{8397378485568779601424058120496725082521164075741015} a - \frac{2432412168137485177363408967521043357239774586474}{31929195762618933845718852169189068754833323481905}$

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

Class group and class number

$C_{10}\times C_{102610}$, which has order $1026100$ (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:  \( 1746210.0427691017 \) (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

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\zeta_{11})^+\), 10.0.21950349414400000.4, 10.10.53339349076992.1, 10.0.5333934907699200000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$