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Properties

Label 20.0.29832890652...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $265.30$
Ramified primes $2, 3, 5, 11$
Class number $303400336$ (GRH)
Class group $[2, 2, 75850084]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![213029036551, -86885984892, 154399084602, -44429897024, 56995864951, -30521105752, 21613556694, -1434041112, 2010972339, -1604074892, 931463700, -150084456, 67801904, -5648720, 2127914, -106884, 34615, -1024, 290, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 290*x^18 - 1024*x^17 + 34615*x^16 - 106884*x^15 + 2127914*x^14 - 5648720*x^13 + 67801904*x^12 - 150084456*x^11 + 931463700*x^10 - 1604074892*x^9 + 2010972339*x^8 - 1434041112*x^7 + 21613556694*x^6 - 30521105752*x^5 + 56995864951*x^4 - 44429897024*x^3 + 154399084602*x^2 - 86885984892*x + 213029036551)
 
gp: K = bnfinit(x^20 - 4*x^19 + 290*x^18 - 1024*x^17 + 34615*x^16 - 106884*x^15 + 2127914*x^14 - 5648720*x^13 + 67801904*x^12 - 150084456*x^11 + 931463700*x^10 - 1604074892*x^9 + 2010972339*x^8 - 1434041112*x^7 + 21613556694*x^6 - 30521105752*x^5 + 56995864951*x^4 - 44429897024*x^3 + 154399084602*x^2 - 86885984892*x + 213029036551, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 290 x^{18} - 1024 x^{17} + 34615 x^{16} - 106884 x^{15} + 2127914 x^{14} - 5648720 x^{13} + 67801904 x^{12} - 150084456 x^{11} + 931463700 x^{10} - 1604074892 x^{9} + 2010972339 x^{8} - 1434041112 x^{7} + 21613556694 x^{6} - 30521105752 x^{5} + 56995864951 x^{4} - 44429897024 x^{3} + 154399084602 x^{2} - 86885984892 x + 213029036551 \)

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:  \(2983289065263288625233938941476864000000000000000=2^{55}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $265.30$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2640=2^{4}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2640}(1,·)$, $\chi_{2640}(323,·)$, $\chi_{2640}(961,·)$, $\chi_{2640}(1609,·)$, $\chi_{2640}(587,·)$, $\chi_{2640}(1681,·)$, $\chi_{2640}(1043,·)$, $\chi_{2640}(1849,·)$, $\chi_{2640}(1307,·)$, $\chi_{2640}(1369,·)$, $\chi_{2640}(2267,·)$, $\chi_{2640}(2401,·)$, $\chi_{2640}(1763,·)$, $\chi_{2640}(169,·)$, $\chi_{2640}(2027,·)$, $\chi_{2640}(2161,·)$, $\chi_{2640}(1523,·)$, $\chi_{2640}(889,·)$, $\chi_{2640}(1787,·)$, $\chi_{2640}(2003,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{9} a^{10} - \frac{2}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{11} + \frac{4}{9} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{12} - \frac{4}{9} a^{9} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{14} + \frac{2}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{15} + \frac{4}{9} a^{9} + \frac{1}{3} a^{8} + \frac{4}{9} a^{7} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{2}{9} a^{9} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{9} a^{6} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{17} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{5210063937533071076658975714993438249} a^{18} - \frac{407577053151138031458162924931798}{1736687979177690358886325238331146083} a^{17} - \frac{8829002574681751624612626780298376}{5210063937533071076658975714993438249} a^{16} - \frac{8060267772832550910025849865266554}{578895993059230119628775079443715361} a^{15} - \frac{3492290944271543139231676910199883}{578895993059230119628775079443715361} a^{14} - \frac{64201853483248018361038866865415996}{5210063937533071076658975714993438249} a^{13} + \frac{57751866916734887855035363502200355}{5210063937533071076658975714993438249} a^{12} + \frac{79636316994272678506726275243980531}{5210063937533071076658975714993438249} a^{11} - \frac{54494886225233921205543446313399116}{1736687979177690358886325238331146083} a^{10} - \frac{253279170317700822204511324140478393}{578895993059230119628775079443715361} a^{9} + \frac{1157896809818868961059973382355433454}{5210063937533071076658975714993438249} a^{8} + \frac{2349448159828863490561592716655383952}{5210063937533071076658975714993438249} a^{7} - \frac{637689348670353966661949483128724684}{5210063937533071076658975714993438249} a^{6} + \frac{529496433612930619257847513709265200}{5210063937533071076658975714993438249} a^{5} + \frac{194835354402668181029289330057203758}{5210063937533071076658975714993438249} a^{4} + \frac{142736062142145104036814537489401284}{5210063937533071076658975714993438249} a^{3} + \frac{1473227735856736164397245433219911026}{5210063937533071076658975714993438249} a^{2} + \frac{800368747258694503832268647755651384}{5210063937533071076658975714993438249} a - \frac{1724342249171992595649243901615533709}{5210063937533071076658975714993438249}$, $\frac{1}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{19} - \frac{2520161029118376785282213920246432}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{18} - \frac{988337690822885373214402858194721250836524405411724367302637631632834}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{17} + \frac{854076117828105680539282377412740395975172687573382955715724445519617}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{16} - \frac{1162072423870234087225414741680255857903378189458239307884999296551888}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{15} + \frac{529501855229788391656575623209272240671082617301141970723321535359523}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{14} - \frac{129769548342587607569584515323734375635418306775501366587745587753248}{3079794955380168571886154979902638610118406896620571208910109976997361} a^{13} + \frac{110137395046019793398118982503064065815634828538781311536787346711578}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{12} - \frac{99344532232394884548340856887756149241184469807677142321899344891403}{3079794955380168571886154979902638610118406896620571208910109976997361} a^{11} + \frac{24047360676506689185181162416269752122773107736420375428400089932749}{3079794955380168571886154979902638610118406896620571208910109976997361} a^{10} + \frac{710803242162421591131744037127348434584727049877441647860195056599656}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{9} - \frac{7313915666398205389750578866392683753640657672107805683333546511362104}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{8} - \frac{3373264376801181221583691579907129030521054270160883525495607119763508}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{7} - \frac{3341513870369803094308313846671725718949849846417208891557264093986129}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{6} + \frac{5096987364032615194723611856902514286538446286653222970013193338953764}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{5} + \frac{10156330006379264051604422002162031065668230366421745412202995679654302}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{4} - \frac{4248028908278736830689273548450342400756234423000303684673600494412408}{27718154598421517146975394819123747491065662069585140880190989792976249} a^{3} - \frac{3446596195831126700540166901771311161821754606476811073012875466799891}{9239384866140505715658464939707915830355220689861713626730329930992083} a^{2} + \frac{5572363951888913684067360502664511316321203567696208128862282300953421}{27718154598421517146975394819123747491065662069585140880190989792976249} a - \frac{2277398875929165467247425608969198793649529649742641796601650990051208}{27718154598421517146975394819123747491065662069585140880190989792976249}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{75850084}$, which has order $303400336$ (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:  \( 14452469.589232503 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{10}) \), 4.0.2304000.1, \(\Q(\zeta_{11})^+\), 10.10.21950349414400000.1

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 R R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
11Data not computed