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Properties

Label 20.0.41278417207...0809.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 31^{18}$
Root discriminant $38.09$
Ramified primes $3, 31$
Class number $15$ (GRH)
Class group $[15]$ (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![625, -500, -1175, -960, 2774, 1617, -2903, -1064, 1792, 1383, 811, -1510, -232, -237, 371, 25, 29, -34, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 - 34*x^17 + 29*x^16 + 25*x^15 + 371*x^14 - 237*x^13 - 232*x^12 - 1510*x^11 + 811*x^10 + 1383*x^9 + 1792*x^8 - 1064*x^7 - 2903*x^6 + 1617*x^5 + 2774*x^4 - 960*x^3 - 1175*x^2 - 500*x + 625)
 
gp: K = bnfinit(x^20 - x^19 - x^18 - 34*x^17 + 29*x^16 + 25*x^15 + 371*x^14 - 237*x^13 - 232*x^12 - 1510*x^11 + 811*x^10 + 1383*x^9 + 1792*x^8 - 1064*x^7 - 2903*x^6 + 1617*x^5 + 2774*x^4 - 960*x^3 - 1175*x^2 - 500*x + 625, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} - 34 x^{17} + 29 x^{16} + 25 x^{15} + 371 x^{14} - 237 x^{13} - 232 x^{12} - 1510 x^{11} + 811 x^{10} + 1383 x^{9} + 1792 x^{8} - 1064 x^{7} - 2903 x^{6} + 1617 x^{5} + 2774 x^{4} - 960 x^{3} - 1175 x^{2} - 500 x + 625 \)

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:  \(41278417207323610480699463560809=3^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.09$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(93=3\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{93}(64,·)$, $\chi_{93}(1,·)$, $\chi_{93}(2,·)$, $\chi_{93}(4,·)$, $\chi_{93}(70,·)$, $\chi_{93}(8,·)$, $\chi_{93}(77,·)$, $\chi_{93}(16,·)$, $\chi_{93}(85,·)$, $\chi_{93}(23,·)$, $\chi_{93}(89,·)$, $\chi_{93}(91,·)$, $\chi_{93}(92,·)$, $\chi_{93}(29,·)$, $\chi_{93}(32,·)$, $\chi_{93}(35,·)$, $\chi_{93}(46,·)$, $\chi_{93}(47,·)$, $\chi_{93}(58,·)$, $\chi_{93}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{1}{5} a$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{16} - \frac{2}{25} a^{15} - \frac{2}{25} a^{13} + \frac{1}{25} a^{12} + \frac{1}{25} a^{11} - \frac{1}{25} a^{10} - \frac{2}{25} a^{9} - \frac{9}{25} a^{8} - \frac{9}{25} a^{7} + \frac{9}{25} a^{6} - \frac{7}{25} a^{5} + \frac{2}{5} a^{4} - \frac{2}{25} a^{3} - \frac{6}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{8375} a^{17} - \frac{162}{8375} a^{16} + \frac{2}{335} a^{15} - \frac{52}{8375} a^{14} + \frac{681}{8375} a^{13} + \frac{321}{8375} a^{12} + \frac{284}{8375} a^{11} - \frac{327}{8375} a^{10} + \frac{496}{8375} a^{9} - \frac{3989}{8375} a^{8} + \frac{3319}{8375} a^{7} - \frac{1957}{8375} a^{6} - \frac{597}{1675} a^{5} + \frac{903}{8375} a^{4} + \frac{104}{8375} a^{3} - \frac{789}{1675} a^{2} + \frac{83}{335} a - \frac{20}{67}$, $\frac{1}{8375} a^{18} - \frac{64}{8375} a^{16} - \frac{662}{8375} a^{15} + \frac{632}{8375} a^{14} - \frac{242}{8375} a^{13} - \frac{309}{8375} a^{12} - \frac{214}{8375} a^{11} + \frac{117}{8375} a^{10} + \frac{653}{8375} a^{9} - \frac{2044}{8375} a^{8} + \frac{2401}{8375} a^{7} + \frac{2251}{8375} a^{6} - \frac{2277}{8375} a^{5} + \frac{468}{1675} a^{4} - \frac{4182}{8375} a^{3} - \frac{639}{1675} a^{2} - \frac{121}{335} a - \frac{24}{67}$, $\frac{1}{450872285142953784125} a^{19} + \frac{21316674166433217}{450872285142953784125} a^{18} - \frac{15559605381363268}{450872285142953784125} a^{17} + \frac{3661215870828319733}{450872285142953784125} a^{16} - \frac{43818508113605840717}{450872285142953784125} a^{15} - \frac{955713489929103203}{90174457028590756825} a^{14} + \frac{14296840225499395608}{450872285142953784125} a^{13} + \frac{8577626063612272309}{450872285142953784125} a^{12} - \frac{25323145523362733672}{450872285142953784125} a^{11} + \frac{7106829809602101798}{90174457028590756825} a^{10} + \frac{12630661596350868553}{450872285142953784125} a^{9} - \frac{167255408799179721181}{450872285142953784125} a^{8} - \frac{149288604813643294698}{450872285142953784125} a^{7} + \frac{69803485119207526508}{450872285142953784125} a^{6} + \frac{196068500505363272451}{450872285142953784125} a^{5} - \frac{28919464576118445689}{450872285142953784125} a^{4} + \frac{695179008050665052}{3606978281143630273} a^{3} + \frac{36777954990158409806}{90174457028590756825} a^{2} - \frac{7391029930605850978}{18034891405718151365} a + \frac{563033300070886669}{3606978281143630273}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  \( \frac{943291821507061566}{450872285142953784125} a^{19} - \frac{103550976585711071}{450872285142953784125} a^{18} - \frac{1361295752455181761}{450872285142953784125} a^{17} - \frac{32976299904816825164}{450872285142953784125} a^{16} - \frac{1697824461414786216}{450872285142953784125} a^{15} + \frac{6610063255029953411}{90174457028590756825} a^{14} + \frac{370773877131182175686}{450872285142953784125} a^{13} + \frac{99640292850664310753}{450872285142953784125} a^{12} - \frac{248347518884265447077}{450872285142953784125} a^{11} - \frac{63219443868674205864}{18034891405718151365} a^{10} - \frac{587509164864807242774}{450872285142953784125} a^{9} + \frac{1261235413371388950473}{450872285142953784125} a^{8} + \frac{2605802275459411045157}{450872285142953784125} a^{7} + \frac{986641841744537147216}{450872285142953784125} a^{6} - \frac{2541858694477372355928}{450872285142953784125} a^{5} - \frac{439942554455999234338}{450872285142953784125} a^{4} + \frac{2902054842772644670349}{450872285142953784125} a^{3} + \frac{315741971728630707274}{90174457028590756825} a^{2} - \frac{27799272823298426647}{18034891405718151365} a - \frac{6248170448456772104}{3606978281143630273} \) (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:  \( 6483708.43992 \) (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

\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{93}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-31})\), 5.5.923521.1, 10.0.26439622160671.1, 10.10.6424828185043053.1, 10.0.207252522098163.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$31$31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.8$x^{10} + 521017$$10$$1$$9$$C_{10}$$[\ ]_{10}$