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Properties

Label 20.0.37404946059...9761.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $42.53$
Ramified primes $3, 11, 13$
Class number $110$ (GRH)
Class group $[110]$ (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![529, -2047, 9324, -16237, 50332, -72035, 188321, -143454, 183245, -61853, 94832, -26202, 26875, -4405, 4546, -700, 479, -49, 30, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 30*x^18 - 49*x^17 + 479*x^16 - 700*x^15 + 4546*x^14 - 4405*x^13 + 26875*x^12 - 26202*x^11 + 94832*x^10 - 61853*x^9 + 183245*x^8 - 143454*x^7 + 188321*x^6 - 72035*x^5 + 50332*x^4 - 16237*x^3 + 9324*x^2 - 2047*x + 529)
 
gp: K = bnfinit(x^20 - 3*x^19 + 30*x^18 - 49*x^17 + 479*x^16 - 700*x^15 + 4546*x^14 - 4405*x^13 + 26875*x^12 - 26202*x^11 + 94832*x^10 - 61853*x^9 + 183245*x^8 - 143454*x^7 + 188321*x^6 - 72035*x^5 + 50332*x^4 - 16237*x^3 + 9324*x^2 - 2047*x + 529, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 30 x^{18} - 49 x^{17} + 479 x^{16} - 700 x^{15} + 4546 x^{14} - 4405 x^{13} + 26875 x^{12} - 26202 x^{11} + 94832 x^{10} - 61853 x^{9} + 183245 x^{8} - 143454 x^{7} + 188321 x^{6} - 72035 x^{5} + 50332 x^{4} - 16237 x^{3} + 9324 x^{2} - 2047 x + 529 \)

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:  \(374049460594333053432589974619761=3^{10}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.53$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(429=3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{429}(64,·)$, $\chi_{429}(1,·)$, $\chi_{429}(196,·)$, $\chi_{429}(389,·)$, $\chi_{429}(14,·)$, $\chi_{429}(25,·)$, $\chi_{429}(155,·)$, $\chi_{429}(92,·)$, $\chi_{429}(157,·)$, $\chi_{429}(350,·)$, $\chi_{429}(287,·)$, $\chi_{429}(38,·)$, $\chi_{429}(103,·)$, $\chi_{429}(298,·)$, $\chi_{429}(235,·)$, $\chi_{429}(53,·)$, $\chi_{429}(311,·)$, $\chi_{429}(313,·)$, $\chi_{429}(170,·)$, $\chi_{429}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{12} - \frac{5}{11} a^{9} - \frac{2}{11} a^{6} + \frac{4}{11} a^{3} - \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{2}{11} a^{13} - \frac{5}{11} a^{10} - \frac{2}{11} a^{7} + \frac{4}{11} a^{4} - \frac{1}{11} a$, $\frac{1}{253} a^{17} - \frac{7}{253} a^{16} - \frac{4}{253} a^{15} + \frac{79}{253} a^{14} - \frac{3}{253} a^{13} + \frac{3}{253} a^{12} + \frac{28}{253} a^{11} + \frac{35}{253} a^{10} - \frac{2}{253} a^{9} + \frac{64}{253} a^{8} + \frac{124}{253} a^{7} + \frac{107}{253} a^{6} - \frac{62}{253} a^{5} + \frac{5}{253} a^{4} + \frac{72}{253} a^{3} - \frac{45}{253} a^{2} + \frac{117}{253} a + \frac{4}{11}$, $\frac{1}{122396593} a^{18} + \frac{54267}{122396593} a^{17} - \frac{646561}{122396593} a^{16} - \frac{1102632}{122396593} a^{15} + \frac{60248966}{122396593} a^{14} + \frac{32460818}{122396593} a^{13} + \frac{19271457}{122396593} a^{12} - \frac{13321020}{122396593} a^{11} - \frac{11219336}{122396593} a^{10} + \frac{11507574}{122396593} a^{9} + \frac{28597572}{122396593} a^{8} - \frac{47015765}{122396593} a^{7} - \frac{21855281}{122396593} a^{6} - \frac{53552846}{122396593} a^{5} + \frac{28577174}{122396593} a^{4} + \frac{49500353}{122396593} a^{3} - \frac{35596345}{122396593} a^{2} + \frac{12701186}{122396593} a + \frac{2550377}{5321591}$, $\frac{1}{152970499338580414732511056113098813591} a^{19} + \frac{1189183333766785564283863929}{323404861180931109371059315249680367} a^{18} - \frac{77131062159028408078527643294974520}{152970499338580414732511056113098813591} a^{17} + \frac{5194241081028357112600781348708716809}{152970499338580414732511056113098813591} a^{16} + \frac{72068327717693978947583525144680528}{3557453472990242203081652467746484037} a^{15} + \frac{7431739769714445613704182922833507919}{152970499338580414732511056113098813591} a^{14} + \frac{2788397861470875795746589334137028018}{6650891275590452814457002439699948417} a^{13} + \frac{55503250730078458440005827135695899639}{152970499338580414732511056113098813591} a^{12} + \frac{67523911782608779948677862460155071494}{152970499338580414732511056113098813591} a^{11} + \frac{17047351272760791936323841012136544305}{152970499338580414732511056113098813591} a^{10} - \frac{1081016206559519859351017602466945866}{3557453472990242203081652467746484037} a^{9} + \frac{76351763022774364284660659063318380431}{152970499338580414732511056113098813591} a^{8} + \frac{51501444290365864838992559421175361361}{152970499338580414732511056113098813591} a^{7} + \frac{59863518412672914460980432140504751358}{152970499338580414732511056113098813591} a^{6} + \frac{10584025890254392055593806226106483852}{152970499338580414732511056113098813591} a^{5} + \frac{25876098174179661021555874579229833946}{152970499338580414732511056113098813591} a^{4} + \frac{2919743615075526459594835842082997473}{152970499338580414732511056113098813591} a^{3} + \frac{36501752402769291957619696949172989469}{152970499338580414732511056113098813591} a^{2} - \frac{2613357022728331214585988700037163947}{152970499338580414732511056113098813591} a + \frac{2768130693557608562053177628126857903}{6650891275590452814457002439699948417}$

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

Class group and class number

$C_{110}$, which has order $110$ (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:  \( -\frac{179720378759538075067534692439}{316197823681749417055467362532011} a^{19} + \frac{12548030233678086492588955826}{7353437760040684117569008430977} a^{18} - \frac{5386891607877618142931013239624}{316197823681749417055467362532011} a^{17} + \frac{8790703978803731188997027121878}{316197823681749417055467362532011} a^{16} - \frac{1997639069745745110703628488451}{7353437760040684117569008430977} a^{15} + \frac{125412756841709073866519603131112}{316197823681749417055467362532011} a^{14} - \frac{73998510109495670562600512302041}{28745256698340856095951578412001} a^{13} + \frac{784688637293657157325132848832979}{316197823681749417055467362532011} a^{12} - \frac{4798190949323326995402287091136261}{316197823681749417055467362532011} a^{11} + \frac{4649999655085882496850484904094374}{316197823681749417055467362532011} a^{10} - \frac{391972061199894100219930446156298}{7353437760040684117569008430977} a^{9} + \frac{10744551509192134933784502210766940}{316197823681749417055467362532011} a^{8} - \frac{32162490085033668709847609726779514}{316197823681749417055467362532011} a^{7} + \frac{24614688872754888608874850738175608}{316197823681749417055467362532011} a^{6} - \frac{32336575265557732863590896484805636}{316197823681749417055467362532011} a^{5} + \frac{10579250179878057361861475012524979}{316197823681749417055467362532011} a^{4} - \frac{6842732435560006665102067623000752}{316197823681749417055467362532011} a^{3} + \frac{941312537024386535235012407050882}{316197823681749417055467362532011} a^{2} - \frac{1198549088481701462790865674974689}{316197823681749417055467362532011} a + \frac{11337083728432291593459133452962}{13747731464423887698063798370957} \) (order $6$)
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:  \( 2015201.7242 \) (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

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\zeta_{11})^+\), 10.0.19340358336761319.3, 10.10.79589952003133.1, 10.0.52089208083.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$13$13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$