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Properties

Label 20.0.44067604925...7169.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 41^{10}$
Root discriminant $95.99$
Ramified primes $3, 11, 41$
Class number $1340130$ (GRH)
Class group $[1340130]$ (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![106391301391, 58008735439, 33729031070, 27761073661, 37048706299, 2210211840, 13956163931, -275733141, 2603211910, -68935889, 273442709, -6541139, 17222530, -344561, 662001, -10240, 15169, -159, 190, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 190*x^18 - 159*x^17 + 15169*x^16 - 10240*x^15 + 662001*x^14 - 344561*x^13 + 17222530*x^12 - 6541139*x^11 + 273442709*x^10 - 68935889*x^9 + 2603211910*x^8 - 275733141*x^7 + 13956163931*x^6 + 2210211840*x^5 + 37048706299*x^4 + 27761073661*x^3 + 33729031070*x^2 + 58008735439*x + 106391301391)
 
gp: K = bnfinit(x^20 - x^19 + 190*x^18 - 159*x^17 + 15169*x^16 - 10240*x^15 + 662001*x^14 - 344561*x^13 + 17222530*x^12 - 6541139*x^11 + 273442709*x^10 - 68935889*x^9 + 2603211910*x^8 - 275733141*x^7 + 13956163931*x^6 + 2210211840*x^5 + 37048706299*x^4 + 27761073661*x^3 + 33729031070*x^2 + 58008735439*x + 106391301391, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 190 x^{18} - 159 x^{17} + 15169 x^{16} - 10240 x^{15} + 662001 x^{14} - 344561 x^{13} + 17222530 x^{12} - 6541139 x^{11} + 273442709 x^{10} - 68935889 x^{9} + 2603211910 x^{8} - 275733141 x^{7} + 13956163931 x^{6} + 2210211840 x^{5} + 37048706299 x^{4} + 27761073661 x^{3} + 33729031070 x^{2} + 58008735439 x + 106391301391 \)

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:  \(4406760492524507138413703843232533837169=3^{10}\cdot 11^{18}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.99$
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:  $3, 11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1353=3\cdot 11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1353}(1,·)$, $\chi_{1353}(901,·)$, $\chi_{1353}(329,·)$, $\chi_{1353}(778,·)$, $\chi_{1353}(206,·)$, $\chi_{1353}(655,·)$, $\chi_{1353}(83,·)$, $\chi_{1353}(983,·)$, $\chi_{1353}(409,·)$, $\chi_{1353}(862,·)$, $\chi_{1353}(614,·)$, $\chi_{1353}(1190,·)$, $\chi_{1353}(40,·)$, $\chi_{1353}(493,·)$, $\chi_{1353}(368,·)$, $\chi_{1353}(245,·)$, $\chi_{1353}(247,·)$, $\chi_{1353}(122,·)$, $\chi_{1353}(124,·)$, $\chi_{1353}(821,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{287891} a^{11} + \frac{110}{287891} a^{9} + \frac{4400}{287891} a^{7} + \frac{77000}{287891} a^{5} - \frac{25782}{287891} a^{3} - \frac{51564}{287891} a - \frac{141863}{287891}$, $\frac{1}{287891} a^{12} + \frac{110}{287891} a^{10} + \frac{4400}{287891} a^{8} + \frac{77000}{287891} a^{6} - \frac{25782}{287891} a^{4} - \frac{51564}{287891} a^{2} - \frac{141863}{287891} a$, $\frac{1}{287891} a^{13} - \frac{7700}{287891} a^{9} - \frac{119109}{287891} a^{7} + \frac{140948}{287891} a^{5} - \frac{94454}{287891} a^{3} - \frac{141863}{287891} a^{2} - \frac{85780}{287891} a + \frac{58816}{287891}$, $\frac{1}{287891} a^{14} - \frac{7700}{287891} a^{10} - \frac{119109}{287891} a^{8} + \frac{140948}{287891} a^{6} - \frac{94454}{287891} a^{4} - \frac{141863}{287891} a^{3} - \frac{85780}{287891} a^{2} + \frac{58816}{287891} a$, $\frac{1}{3166801} a^{15} - \frac{4}{3166801} a^{13} - \frac{1}{3166801} a^{12} + \frac{2}{3166801} a^{11} - \frac{288001}{3166801} a^{10} + \frac{183129}{3166801} a^{9} - \frac{292291}{3166801} a^{8} - \frac{1192300}{3166801} a^{7} - \frac{364891}{3166801} a^{6} + \frac{1355531}{3166801} a^{5} + \frac{1817}{12517} a^{4} + \frac{1227535}{3166801} a^{3} - \frac{473732}{3166801} a^{2} - \frac{1098711}{3166801} a + \frac{11219}{137687}$, $\frac{1}{891923842611447811} a^{16} + \frac{100474820107}{891923842611447811} a^{15} - \frac{606284029300}{891923842611447811} a^{14} - \frac{172028091431}{891923842611447811} a^{13} - \frac{3056279}{81083985691949801} a^{12} - \frac{673061770024}{891923842611447811} a^{11} + \frac{276493526121803466}{891923842611447811} a^{10} - \frac{61268751349029562}{891923842611447811} a^{9} + \frac{421499761772067982}{891923842611447811} a^{8} - \frac{112071946256979057}{891923842611447811} a^{7} + \frac{349115225258174642}{891923842611447811} a^{6} - \frac{16504567975330719}{38779297504845557} a^{5} - \frac{238107488140850393}{891923842611447811} a^{4} + \frac{139093765604574414}{891923842611447811} a^{3} - \frac{52053528589850292}{891923842611447811} a^{2} + \frac{60592624763993336}{891923842611447811} a - \frac{262017723674075283}{891923842611447811}$, $\frac{1}{891923842611447811} a^{17} - \frac{103956355192}{891923842611447811} a^{15} - \frac{247400608093}{891923842611447811} a^{14} - \frac{1063671803110}{891923842611447811} a^{13} + \frac{818926690961}{891923842611447811} a^{12} + \frac{665222289912}{891923842611447811} a^{11} + \frac{258126534762108779}{891923842611447811} a^{10} + \frac{79372429203103275}{891923842611447811} a^{9} - \frac{292416849944235657}{891923842611447811} a^{8} - \frac{137513138247777548}{891923842611447811} a^{7} - \frac{95124131152592436}{891923842611447811} a^{6} + \frac{432517493702928455}{891923842611447811} a^{5} - \frac{401131207554562975}{891923842611447811} a^{4} + \frac{8333136010259795}{891923842611447811} a^{3} - \frac{352394208805443694}{891923842611447811} a^{2} - \frac{55271141181674481}{891923842611447811} a + \frac{284503609899895984}{891923842611447811}$, $\frac{1}{891923842611447811} a^{18} + \frac{123061187253}{891923842611447811} a^{15} + \frac{696563278265}{891923842611447811} a^{14} - \frac{1287888629402}{891923842611447811} a^{13} + \frac{1132644830464}{891923842611447811} a^{12} + \frac{1478794454306}{891923842611447811} a^{11} - \frac{445541205440938542}{891923842611447811} a^{10} - \frac{22676955479751618}{891923842611447811} a^{9} - \frac{8280997318170677}{38779297504845557} a^{8} - \frac{211766349974852477}{891923842611447811} a^{7} - \frac{115917291591943579}{891923842611447811} a^{6} + \frac{220313012106917542}{891923842611447811} a^{5} - \frac{33470510258992163}{891923842611447811} a^{4} - \frac{98277335750300199}{891923842611447811} a^{3} - \frac{2738147875702137}{81083985691949801} a^{2} + \frac{191692582693313191}{891923842611447811} a - \frac{213135887218705005}{891923842611447811}$, $\frac{1}{891923842611447811} a^{19} + \frac{3302962810}{81083985691949801} a^{15} - \frac{732974921213}{891923842611447811} a^{14} - \frac{970718325031}{891923842611447811} a^{13} - \frac{368475406119}{891923842611447811} a^{12} + \frac{1476070913714}{891923842611447811} a^{11} - \frac{171152128934876102}{891923842611447811} a^{10} + \frac{444597394928210269}{891923842611447811} a^{9} - \frac{99210050705702005}{891923842611447811} a^{8} + \frac{109796180276986685}{891923842611447811} a^{7} - \frac{321549634320153439}{891923842611447811} a^{6} - \frac{186125695352092476}{891923842611447811} a^{5} - \frac{224670498411949086}{891923842611447811} a^{4} - \frac{108599289237989578}{891923842611447811} a^{3} - \frac{379474361481891872}{891923842611447811} a^{2} - \frac{57914315124748781}{891923842611447811} a + \frac{321348208612586328}{891923842611447811}$

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

Class group and class number

$C_{1340130}$, which has order $1340130$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-123}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-451}) \), \(\Q(\sqrt{33}, \sqrt{-123})\), \(\Q(\zeta_{11})^+\), 10.0.6034857761594872683.1, \(\Q(\zeta_{33})^+\), 10.0.273182861635981891.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.10.5.2$x^{10} - 2825761 x^{2} + 810993407$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41.10.5.2$x^{10} - 2825761 x^{2} + 810993407$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$