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Properties

Label 20.0.18589228510...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{16}\cdot 23^{10}$
Root discriminant $73.02$
Ramified primes $5, 11, 23$
Class number $77568$ (GRH)
Class group $[4, 4, 4, 1212]$ (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![1232556799, -975548791, 1365577686, -811477632, 638638334, -301380451, 172058932, -67154201, 30705792, -10220558, 3936196, -1137389, 377005, -94767, 27289, -5990, 1519, -288, 61, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 61*x^18 - 288*x^17 + 1519*x^16 - 5990*x^15 + 27289*x^14 - 94767*x^13 + 377005*x^12 - 1137389*x^11 + 3936196*x^10 - 10220558*x^9 + 30705792*x^8 - 67154201*x^7 + 172058932*x^6 - 301380451*x^5 + 638638334*x^4 - 811477632*x^3 + 1365577686*x^2 - 975548791*x + 1232556799)
 
gp: K = bnfinit(x^20 - 8*x^19 + 61*x^18 - 288*x^17 + 1519*x^16 - 5990*x^15 + 27289*x^14 - 94767*x^13 + 377005*x^12 - 1137389*x^11 + 3936196*x^10 - 10220558*x^9 + 30705792*x^8 - 67154201*x^7 + 172058932*x^6 - 301380451*x^5 + 638638334*x^4 - 811477632*x^3 + 1365577686*x^2 - 975548791*x + 1232556799, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 61 x^{18} - 288 x^{17} + 1519 x^{16} - 5990 x^{15} + 27289 x^{14} - 94767 x^{13} + 377005 x^{12} - 1137389 x^{11} + 3936196 x^{10} - 10220558 x^{9} + 30705792 x^{8} - 67154201 x^{7} + 172058932 x^{6} - 301380451 x^{5} + 638638334 x^{4} - 811477632 x^{3} + 1365577686 x^{2} - 975548791 x + 1232556799 \)

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:  \(18589228510326313111158199467666015625=5^{10}\cdot 11^{16}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.02$
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:  $5, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1265=5\cdot 11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1265}(896,·)$, $\chi_{1265}(1,·)$, $\chi_{1265}(599,·)$, $\chi_{1265}(344,·)$, $\chi_{1265}(1241,·)$, $\chi_{1265}(346,·)$, $\chi_{1265}(91,·)$, $\chi_{1265}(1126,·)$, $\chi_{1265}(1059,·)$, $\chi_{1265}(804,·)$, $\chi_{1265}(229,·)$, $\chi_{1265}(806,·)$, $\chi_{1265}(551,·)$, $\chi_{1265}(829,·)$, $\chi_{1265}(944,·)$, $\chi_{1265}(114,·)$, $\chi_{1265}(691,·)$, $\chi_{1265}(576,·)$, $\chi_{1265}(1149,·)$, $\chi_{1265}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} + \frac{122}{331} a^{17} - \frac{143}{331} a^{16} + \frac{32}{331} a^{15} + \frac{64}{331} a^{14} + \frac{8}{331} a^{13} - \frac{147}{331} a^{12} - \frac{97}{331} a^{11} + \frac{18}{331} a^{10} + \frac{141}{331} a^{9} - \frac{120}{331} a^{8} + \frac{26}{331} a^{7} - \frac{79}{331} a^{6} + \frac{121}{331} a^{5} + \frac{11}{331} a^{4} + \frac{8}{331} a^{3} + \frac{80}{331} a^{2} + \frac{37}{331} a + \frac{112}{331}$, $\frac{1}{68730702715295126985580035853021163092856651588239201320522237788353} a^{19} - \frac{4213499342842953802070148193575358278716800702424987052308445759}{68730702715295126985580035853021163092856651588239201320522237788353} a^{18} - \frac{86753832483936124532561691821905000268835650888227058533493646669}{207645627538655972766102827350517109041862995734861635409432742563} a^{17} + \frac{10659698462395752963448653801688051061587260538893370451218238093040}{68730702715295126985580035853021163092856651588239201320522237788353} a^{16} - \frac{34039127216373573441520368036441000473036140455826570895331682342507}{68730702715295126985580035853021163092856651588239201320522237788353} a^{15} + \frac{30132209091745295006445874092640523812790223791207357010154802304910}{68730702715295126985580035853021163092856651588239201320522237788353} a^{14} + \frac{5372496762248423308200846276712843849979089369422075038949176693685}{68730702715295126985580035853021163092856651588239201320522237788353} a^{13} + \frac{4621658286076578759313678325542949477252581071532712294619634263352}{68730702715295126985580035853021163092856651588239201320522237788353} a^{12} - \frac{13660759281242147979018375927483267269195495013621994132511247647724}{68730702715295126985580035853021163092856651588239201320522237788353} a^{11} + \frac{2728644223452521268420046384429430532891754272681064447764463522809}{68730702715295126985580035853021163092856651588239201320522237788353} a^{10} - \frac{25489000150584516899165751504914480531207025568339413959985916615323}{68730702715295126985580035853021163092856651588239201320522237788353} a^{9} + \frac{1542620131276330474289354489279118569868037526803645679168433345596}{68730702715295126985580035853021163092856651588239201320522237788353} a^{8} + \frac{27608085987129348343018506528868382843187020592526182783066024459182}{68730702715295126985580035853021163092856651588239201320522237788353} a^{7} - \frac{23678404680780217464137035955059010541932303155795803891582917618711}{68730702715295126985580035853021163092856651588239201320522237788353} a^{6} + \frac{31325660558580123611411191601128759651167692812590913007703269999454}{68730702715295126985580035853021163092856651588239201320522237788353} a^{5} - \frac{7908968647086553030548964660068048595459910552220531955185148343490}{68730702715295126985580035853021163092856651588239201320522237788353} a^{4} - \frac{6288019580548288304705661904418642604245927571939512312226842090491}{68730702715295126985580035853021163092856651588239201320522237788353} a^{3} + \frac{27870561158924775675764959930285350394627466672027993537325099739157}{68730702715295126985580035853021163092856651588239201320522237788353} a^{2} + \frac{30544196477654561952785982801078366213144145641723841470800455643987}{68730702715295126985580035853021163092856651588239201320522237788353} a + \frac{24187675922378739516869728304649629683416246186211356190648238869368}{68730702715295126985580035853021163092856651588239201320522237788353}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{1212}$, which has order $77568$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.1379687283212183.1, 10.0.4311522760038071875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$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}$
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$