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Properties

Label 20.0.16468295313...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $144.82$
Ramified primes $2, 5, 7, 11$
Class number $7288160$ (GRH)
Class group $[2, 2, 1822040]$ (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![136721229899, -49883642958, 60171906331, -22011788834, 14745885715, -4925515332, 2409015362, -736111220, 288169059, -79062874, 26069517, -6416014, 1800057, -388732, 94434, -17612, 3553, -534, 91, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 91*x^18 - 534*x^17 + 3553*x^16 - 17612*x^15 + 94434*x^14 - 388732*x^13 + 1800057*x^12 - 6416014*x^11 + 26069517*x^10 - 79062874*x^9 + 288169059*x^8 - 736111220*x^7 + 2409015362*x^6 - 4925515332*x^5 + 14745885715*x^4 - 22011788834*x^3 + 60171906331*x^2 - 49883642958*x + 136721229899)
 
gp: K = bnfinit(x^20 - 10*x^19 + 91*x^18 - 534*x^17 + 3553*x^16 - 17612*x^15 + 94434*x^14 - 388732*x^13 + 1800057*x^12 - 6416014*x^11 + 26069517*x^10 - 79062874*x^9 + 288169059*x^8 - 736111220*x^7 + 2409015362*x^6 - 4925515332*x^5 + 14745885715*x^4 - 22011788834*x^3 + 60171906331*x^2 - 49883642958*x + 136721229899, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 91 x^{18} - 534 x^{17} + 3553 x^{16} - 17612 x^{15} + 94434 x^{14} - 388732 x^{13} + 1800057 x^{12} - 6416014 x^{11} + 26069517 x^{10} - 79062874 x^{9} + 288169059 x^{8} - 736111220 x^{7} + 2409015362 x^{6} - 4925515332 x^{5} + 14745885715 x^{4} - 22011788834 x^{3} + 60171906331 x^{2} - 49883642958 x + 136721229899 \)

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:  \(16468295313503070686136120314429440000000000=2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.82$
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$$\chi_{3080}(1,·)$, $\chi_{3080}(1539,·)$, $\chi_{3080}(1609,·)$, $\chi_{3080}(139,·)$, $\chi_{3080}(2729,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(1891,·)$, $\chi_{3080}(1049,·)$, $\chi_{3080}(1051,·)$, $\chi_{3080}(2659,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(491,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(1329,·)$, $\chi_{3080}(211,·)$, $\chi_{3080}(489,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2939,·)$, $\chi_{3080}(699,·)$, $\chi_{3080}(3011,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{184} a^{14} - \frac{7}{184} a^{13} + \frac{9}{184} a^{12} - \frac{9}{184} a^{11} + \frac{5}{92} a^{10} - \frac{1}{46} a^{9} - \frac{1}{8} a^{8} + \frac{27}{184} a^{7} + \frac{3}{92} a^{6} - \frac{1}{46} a^{5} + \frac{41}{184} a^{4} - \frac{41}{184} a^{3} - \frac{87}{184} a^{2} - \frac{57}{184} a + \frac{7}{46}$, $\frac{1}{184} a^{15} + \frac{3}{92} a^{13} + \frac{1}{23} a^{12} - \frac{7}{184} a^{11} + \frac{5}{46} a^{10} - \frac{5}{184} a^{9} + \frac{1}{46} a^{8} - \frac{35}{184} a^{7} - \frac{1}{23} a^{6} - \frac{33}{184} a^{5} + \frac{2}{23} a^{4} - \frac{13}{46} a^{3} + \frac{3}{23} a^{2} - \frac{3}{184} a + \frac{3}{46}$, $\frac{1}{368} a^{16} + \frac{1}{92} a^{13} + \frac{1}{46} a^{12} + \frac{7}{92} a^{11} - \frac{21}{184} a^{10} + \frac{7}{92} a^{9} + \frac{11}{368} a^{8} + \frac{15}{92} a^{7} + \frac{1}{8} a^{6} - \frac{13}{92} a^{5} - \frac{11}{184} a^{4} - \frac{13}{92} a^{3} - \frac{5}{184} a^{2} - \frac{19}{46} a - \frac{53}{368}$, $\frac{1}{368} a^{17} - \frac{5}{184} a^{13} - \frac{1}{46} a^{12} + \frac{5}{46} a^{11} - \frac{3}{92} a^{10} + \frac{27}{368} a^{9} - \frac{2}{23} a^{8} - \frac{1}{23} a^{7} - \frac{19}{92} a^{6} - \frac{3}{184} a^{5} - \frac{2}{23} a^{4} - \frac{21}{46} a^{3} - \frac{43}{92} a^{2} - \frac{147}{368} a + \frac{9}{46}$, $\frac{1}{13779286807877387618825155899664} a^{18} - \frac{9}{13779286807877387618825155899664} a^{17} - \frac{7968831315049011116200001075}{6889643403938693809412577949832} a^{16} - \frac{32374354021190016048071826}{20028033150984575027362145203} a^{15} + \frac{10483265882209574213942525303}{6889643403938693809412577949832} a^{14} + \frac{84067035880256149886726356619}{6889643403938693809412577949832} a^{13} - \frac{273204636735407493135640793979}{6889643403938693809412577949832} a^{12} - \frac{148680231268558832961816658603}{3444821701969346904706288974916} a^{11} + \frac{17181833639867401852644562759}{320448530415753200437794323248} a^{10} - \frac{28872168726893020623048098101}{320448530415753200437794323248} a^{9} - \frac{356928913900951857753560198457}{6889643403938693809412577949832} a^{8} + \frac{16032703264886321080609842899}{3444821701969346904706288974916} a^{7} - \frac{682755836682117147850244361961}{3444821701969346904706288974916} a^{6} - \frac{1201270137349728168968306070159}{6889643403938693809412577949832} a^{5} + \frac{188575039074863867198121534501}{3444821701969346904706288974916} a^{4} + \frac{1708123716791379884530796595617}{3444821701969346904706288974916} a^{3} - \frac{4000325275532612048178007328417}{13779286807877387618825155899664} a^{2} + \frac{2940871827160003087601274149643}{13779286807877387618825155899664} a - \frac{1298458794125179547418612546775}{3444821701969346904706288974916}$, $\frac{1}{4986069696571048984520844958161753252272} a^{19} + \frac{45231563}{1246517424142762246130211239540438313068} a^{18} - \frac{3261173030855042393612607434121387319}{4986069696571048984520844958161753252272} a^{17} - \frac{1130887913556388338218201035531115617}{2493034848285524492260422479080876626136} a^{16} + \frac{276932332783244770081607302344070923}{2493034848285524492260422479080876626136} a^{15} - \frac{1186316404136274011446926464764548159}{2493034848285524492260422479080876626136} a^{14} + \frac{11880986124491305680037614208699736553}{1246517424142762246130211239540438313068} a^{13} + \frac{3543332342467398402121611797455405413}{311629356035690561532552809885109578267} a^{12} - \frac{222224745258008903449701535258116583219}{4986069696571048984520844958161753252272} a^{11} - \frac{4321037990377846701700740583695228387}{57977554611291267261870290211183177352} a^{10} - \frac{528162246512469140683463688135511577429}{4986069696571048984520844958161753252272} a^{9} - \frac{1674081754183426101319376041549250293}{54196409745337488962183097371323404916} a^{8} + \frac{110687697284867944740933138418427190665}{1246517424142762246130211239540438313068} a^{7} - \frac{603594908587419249017536227978480396979}{2493034848285524492260422479080876626136} a^{6} + \frac{233298165584154804135477099485461261813}{2493034848285524492260422479080876626136} a^{5} + \frac{58330516609260955701308329394509574627}{2493034848285524492260422479080876626136} a^{4} - \frac{98970991353840963205933333246536671297}{4986069696571048984520844958161753252272} a^{3} - \frac{27382103209930792789803642721919851141}{1246517424142762246130211239540438313068} a^{2} - \frac{2195072056005854047613752540780605374617}{4986069696571048984520844958161753252272} a + \frac{313468213575934707299447209002262619371}{1246517424142762246130211239540438313068}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1822040}$, which has order $7288160$ (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:  \( 1589230.0087159988 \) (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

\(\Q(\sqrt{-770}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{22}, \sqrt{-35})\), \(\Q(\zeta_{11})^+\), 10.0.4058114748686028800000.1, 10.0.11258530353021875.4, 10.10.77265229938688.1

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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$