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Properties

Label 20.0.16863534401...000.12
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $204.81$
Ramified primes $2, 5, 7, 11$
Class number $64135808$ (GRH)
Class group $[2, 88, 364408]$ (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![65144737570441, 0, 8871244946010, 0, 863742291395, 0, 53381523010, 0, 2949712395, 0, 110497408, 0, 4425040, 0, 96630, 0, 3265, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 30*x^18 + 3265*x^16 + 96630*x^14 + 4425040*x^12 + 110497408*x^10 + 2949712395*x^8 + 53381523010*x^6 + 863742291395*x^4 + 8871244946010*x^2 + 65144737570441)
 
gp: K = bnfinit(x^20 + 30*x^18 + 3265*x^16 + 96630*x^14 + 4425040*x^12 + 110497408*x^10 + 2949712395*x^8 + 53381523010*x^6 + 863742291395*x^4 + 8871244946010*x^2 + 65144737570441, 1)
 

Normalized defining polynomial

\( x^{20} + 30 x^{18} + 3265 x^{16} + 96630 x^{14} + 4425040 x^{12} + 110497408 x^{10} + 2949712395 x^{8} + 53381523010 x^{6} + 863742291395 x^{4} + 8871244946010 x^{2} + 65144737570441 \)

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:  \(16863534401027144382603387201975746560000000000=2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.81$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1539,·)$, $\chi_{3080}(699,·)$, $\chi_{3080}(519,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(139,·)$, $\chi_{3080}(1359,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(1301,·)$, $\chi_{3080}(2199,·)$, $\chi_{3080}(799,·)$, $\chi_{3080}(2659,·)$, $\chi_{3080}(741,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(239,·)$, $\chi_{3080}(181,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2939,·)$, $\chi_{3080}(1021,·)$, $\chi_{3080}(2421,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{6} + \frac{1}{25} a^{4} - \frac{9}{25} a^{2} + \frac{6}{25}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{7} + \frac{1}{25} a^{5} - \frac{9}{25} a^{3} + \frac{6}{25} a$, $\frac{1}{50} a^{10} - \frac{1}{50} a^{8} + \frac{2}{25} a^{6} - \frac{1}{50} a^{4} - \frac{11}{50} a^{2} - \frac{17}{50}$, $\frac{1}{50} a^{11} - \frac{1}{50} a^{9} + \frac{2}{25} a^{7} - \frac{1}{50} a^{5} - \frac{11}{50} a^{3} - \frac{17}{50} a$, $\frac{1}{250} a^{12} - \frac{1}{250} a^{10} + \frac{3}{50} a^{6} - \frac{1}{10} a^{4} - \frac{121}{250} a^{2} + \frac{3}{125}$, $\frac{1}{250} a^{13} - \frac{1}{250} a^{11} + \frac{3}{50} a^{7} - \frac{1}{10} a^{5} - \frac{121}{250} a^{3} + \frac{3}{125} a$, $\frac{1}{250} a^{14} - \frac{1}{250} a^{10} - \frac{1}{50} a^{8} + \frac{2}{25} a^{6} - \frac{8}{125} a^{4} + \frac{23}{50} a^{2} - \frac{57}{125}$, $\frac{1}{11500} a^{15} - \frac{1}{500} a^{14} - \frac{11}{11500} a^{13} + \frac{1}{2300} a^{11} - \frac{1}{125} a^{10} + \frac{6}{575} a^{9} - \frac{1}{50} a^{8} - \frac{219}{2300} a^{7} - \frac{1}{50} a^{6} - \frac{433}{5750} a^{5} - \frac{49}{500} a^{4} - \frac{929}{11500} a^{3} + \frac{7}{50} a^{2} + \frac{61}{460} a + \frac{129}{500}$, $\frac{1}{57500} a^{16} + \frac{3}{14375} a^{14} - \frac{1}{500} a^{13} - \frac{87}{57500} a^{12} + \frac{1}{500} a^{11} - \frac{193}{28750} a^{10} + \frac{149}{11500} a^{8} - \frac{3}{100} a^{7} - \frac{964}{14375} a^{6} + \frac{1}{20} a^{5} + \frac{1027}{14375} a^{4} + \frac{121}{500} a^{3} - \frac{24143}{57500} a^{2} + \frac{61}{125} a - \frac{903}{2500}$, $\frac{1}{57500} a^{17} + \frac{1}{28750} a^{15} - \frac{1}{500} a^{14} + \frac{1}{2500} a^{13} - \frac{1}{500} a^{12} - \frac{109}{14375} a^{11} + \frac{1}{250} a^{10} - \frac{91}{11500} a^{9} + \frac{1}{100} a^{8} - \frac{2203}{28750} a^{7} + \frac{3}{100} a^{6} + \frac{317}{14375} a^{5} - \frac{9}{500} a^{4} - \frac{14853}{57500} a^{3} + \frac{103}{250} a^{2} - \frac{13019}{57500} a - \frac{23}{125}$, $\frac{1}{3504557757495352304319063724440237417500} a^{18} - \frac{1442036396545252734641294949215037}{700911551499070460863812744888047483500} a^{16} - \frac{1581449396359431381766850330424253189}{876139439373838076079765931110059354375} a^{14} - \frac{1}{500} a^{13} - \frac{3466025911035167325470000798401819357}{3504557757495352304319063724440237417500} a^{12} - \frac{1}{125} a^{11} - \frac{13173280566096004778343833248945453173}{3504557757495352304319063724440237417500} a^{10} + \frac{1}{100} a^{9} + \frac{34141425897986106642899389565938069879}{3504557757495352304319063724440237417500} a^{8} - \frac{7}{100} a^{7} + \frac{8566348684005189280734935129972170667}{175227887874767615215953186222011870875} a^{6} + \frac{3}{50} a^{5} + \frac{4315795294475653863754595788138436984}{876139439373838076079765931110059354375} a^{4} - \frac{37}{250} a^{3} + \frac{218738344420674781284178076620241514028}{876139439373838076079765931110059354375} a^{2} - \frac{171}{500} a - \frac{13635737857068153095697226036532992866}{38093019103210351133902866570002580625}$, $\frac{1}{1229829921933541516688558799371741435261352500} a^{19} + \frac{29015271438110980712234585029005701351}{4919319687734166066754235197486965741045410} a^{17} - \frac{14470397523150820892474521449417956886391}{1229829921933541516688558799371741435261352500} a^{15} - \frac{1}{500} a^{14} + \frac{57245057930338422827716326120625624104681}{53470866171023544203850382581380062402667500} a^{13} - \frac{1}{500} a^{12} + \frac{116701991192335361118840152670233820098371}{13367716542755886050962595645345015600666875} a^{11} - \frac{3}{500} a^{10} - \frac{3616883630823289972768725857964455043116624}{307457480483385379172139699842935358815338125} a^{9} - \frac{1}{50} a^{8} - \frac{12475961147555673661824771517463469478022099}{245965984386708303337711759874348287052270500} a^{7} - \frac{1}{20} a^{6} - \frac{5733602414334444955836996611494192366771001}{307457480483385379172139699842935358815338125} a^{5} - \frac{6}{125} a^{4} + \frac{135537222798134934909711269699016886803606173}{307457480483385379172139699842935358815338125} a^{3} + \frac{191}{500} a^{2} - \frac{117948649303326609171190850961377094470391963}{307457480483385379172139699842935358815338125} a - \frac{127}{500}$

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

Class group and class number

$C_{2}\times C_{88}\times C_{364408}$, which has order $64135808$ (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:  \( 11184526.893275889 \) (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

\(\Q(\sqrt{-770}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-14}, \sqrt{55})\), \(\Q(\zeta_{11})^+\), 10.0.4058114748686028800000.1, 10.0.118054247234502656.1, 10.10.7545432611200000.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
11Data not computed