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Properties

Label 20.0.36905625000...000.10
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{34}$
Root discriminant $75.57$
Ramified primes $2, 3, 5$
Class number $264620$ (GRH)
Class group $[264620]$ (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![914911255, -16634750, 709001375, -2516900, 274337325, 220760, 69614025, 119100, 12766770, 20230, 1772491, 1500, 189475, 90, 15520, -2, 950, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 40*x^18 + 950*x^16 - 2*x^15 + 15520*x^14 + 90*x^13 + 189475*x^12 + 1500*x^11 + 1772491*x^10 + 20230*x^9 + 12766770*x^8 + 119100*x^7 + 69614025*x^6 + 220760*x^5 + 274337325*x^4 - 2516900*x^3 + 709001375*x^2 - 16634750*x + 914911255)
 
gp: K = bnfinit(x^20 + 40*x^18 + 950*x^16 - 2*x^15 + 15520*x^14 + 90*x^13 + 189475*x^12 + 1500*x^11 + 1772491*x^10 + 20230*x^9 + 12766770*x^8 + 119100*x^7 + 69614025*x^6 + 220760*x^5 + 274337325*x^4 - 2516900*x^3 + 709001375*x^2 - 16634750*x + 914911255, 1)
 

Normalized defining polynomial

\( x^{20} + 40 x^{18} + 950 x^{16} - 2 x^{15} + 15520 x^{14} + 90 x^{13} + 189475 x^{12} + 1500 x^{11} + 1772491 x^{10} + 20230 x^{9} + 12766770 x^{8} + 119100 x^{7} + 69614025 x^{6} + 220760 x^{5} + 274337325 x^{4} - 2516900 x^{3} + 709001375 x^{2} - 16634750 x + 914911255 \)

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:  \(36905625000000000000000000000000000000=2^{30}\cdot 3^{10}\cdot 5^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.57$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(600=2^{3}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{600}(1,·)$, $\chi_{600}(389,·)$, $\chi_{600}(481,·)$, $\chi_{600}(269,·)$, $\chi_{600}(461,·)$, $\chi_{600}(529,·)$, $\chi_{600}(341,·)$, $\chi_{600}(409,·)$, $\chi_{600}(221,·)$, $\chi_{600}(581,·)$, $\chi_{600}(289,·)$, $\chi_{600}(101,·)$, $\chi_{600}(49,·)$, $\chi_{600}(169,·)$, $\chi_{600}(29,·)$, $\chi_{600}(241,·)$, $\chi_{600}(361,·)$, $\chi_{600}(121,·)$, $\chi_{600}(509,·)$, $\chi_{600}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{707} a^{18} + \frac{249}{707} a^{17} + \frac{142}{707} a^{16} - \frac{243}{707} a^{15} + \frac{304}{707} a^{14} + \frac{11}{101} a^{13} + \frac{38}{101} a^{12} + \frac{237}{707} a^{11} - \frac{187}{707} a^{10} - \frac{335}{707} a^{9} + \frac{162}{707} a^{8} + \frac{328}{707} a^{7} - \frac{18}{101} a^{6} + \frac{15}{101} a^{5} + \frac{320}{707} a^{4} + \frac{3}{101} a^{3} + \frac{27}{707} a^{2} - \frac{33}{707} a + \frac{200}{707}$, $\frac{1}{225750894272440855807356226577426449684481122644184148523686407} a^{19} - \frac{13968704660097306320591011701749240739247238482002764026708}{225750894272440855807356226577426449684481122644184148523686407} a^{18} - \frac{49945539324334387902769757828304640400698071887598914001157709}{225750894272440855807356226577426449684481122644184148523686407} a^{17} + \frac{80379203869197818763648007801956007497896967400296907277253911}{225750894272440855807356226577426449684481122644184148523686407} a^{16} - \frac{40738973209809071773041115280606811948774837047869340730149922}{225750894272440855807356226577426449684481122644184148523686407} a^{15} + \frac{7497391041032923575063981604153188039450967744315100632502776}{32250127753205836543908032368203778526354446092026306931955201} a^{14} + \frac{11854108272333416732521796765109405284490969238303788620644821}{32250127753205836543908032368203778526354446092026306931955201} a^{13} - \frac{17960173439289111726225109573268594365555415819917188967711986}{225750894272440855807356226577426449684481122644184148523686407} a^{12} + \frac{41736065424248405526524040046429541598139688598685650013705324}{225750894272440855807356226577426449684481122644184148523686407} a^{11} + \frac{52829683792223101634541828103306063999706549341151662232260358}{225750894272440855807356226577426449684481122644184148523686407} a^{10} + \frac{41159514586091573717983465270283741183549561572694844347324684}{225750894272440855807356226577426449684481122644184148523686407} a^{9} - \frac{6128140616570217321519420298727731144996366492090512676947359}{225750894272440855807356226577426449684481122644184148523686407} a^{8} + \frac{14406099735157770051834284106654963978028007858390132779048119}{32250127753205836543908032368203778526354446092026306931955201} a^{7} + \frac{6407501525950393001743149826857222597215892770052804638923971}{32250127753205836543908032368203778526354446092026306931955201} a^{6} - \frac{53651814246058330125561985536391196487962843810932358695252942}{225750894272440855807356226577426449684481122644184148523686407} a^{5} + \frac{12908674663098709920240204569458199690120966804963926598673506}{32250127753205836543908032368203778526354446092026306931955201} a^{4} + \frac{12949669692191603604108493800583741701234660966932123787934818}{225750894272440855807356226577426449684481122644184148523686407} a^{3} + \frac{98832942874059510591065167908389765976294235381771205019610548}{225750894272440855807356226577426449684481122644184148523686407} a^{2} - \frac{109209698901821592495381801639946690567254618323628190791527349}{225750894272440855807356226577426449684481122644184148523686407} a + \frac{4823043612422350792174670338788600420145901159057069709498428}{32250127753205836543908032368203778526354446092026306931955201}$

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

Class group and class number

$C_{264620}$, which has order $264620$ (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:  \( 161406.837641 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{5}, \sqrt{-6})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.1215000000000000000.45, 10.0.6075000000000000000.26

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
$5$5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$