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Properties

Label 20.0.78487737921...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{18}\cdot 13^{10}$
Root discriminant $139.55$
Ramified primes $2, 5, 11, 13$
Class number $1562000$ (GRH)
Class group $[10, 156200]$ (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![165766539709, -40033619716, 103548492952, -91720108968, 25037292164, 5958300606, -2332001357, -985657428, 498877289, 57316392, -26650514, -4482840, 2172498, 137844, -66026, -5792, 2854, 84, -41, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 41*x^18 + 84*x^17 + 2854*x^16 - 5792*x^15 - 66026*x^14 + 137844*x^13 + 2172498*x^12 - 4482840*x^11 - 26650514*x^10 + 57316392*x^9 + 498877289*x^8 - 985657428*x^7 - 2332001357*x^6 + 5958300606*x^5 + 25037292164*x^4 - 91720108968*x^3 + 103548492952*x^2 - 40033619716*x + 165766539709)
 
gp: K = bnfinit(x^20 - 2*x^19 - 41*x^18 + 84*x^17 + 2854*x^16 - 5792*x^15 - 66026*x^14 + 137844*x^13 + 2172498*x^12 - 4482840*x^11 - 26650514*x^10 + 57316392*x^9 + 498877289*x^8 - 985657428*x^7 - 2332001357*x^6 + 5958300606*x^5 + 25037292164*x^4 - 91720108968*x^3 + 103548492952*x^2 - 40033619716*x + 165766539709, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 41 x^{18} + 84 x^{17} + 2854 x^{16} - 5792 x^{15} - 66026 x^{14} + 137844 x^{13} + 2172498 x^{12} - 4482840 x^{11} - 26650514 x^{10} + 57316392 x^{9} + 498877289 x^{8} - 985657428 x^{7} - 2332001357 x^{6} + 5958300606 x^{5} + 25037292164 x^{4} - 91720108968 x^{3} + 103548492952 x^{2} - 40033619716 x + 165766539709 \)

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:  \(7848773792186191578606179311298560000000000=2^{20}\cdot 5^{10}\cdot 11^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.55$
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:  $2, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2860=2^{2}\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2860}(1,·)$, $\chi_{2860}(1481,·)$, $\chi_{2860}(779,·)$, $\chi_{2860}(79,·)$, $\chi_{2860}(2001,·)$, $\chi_{2860}(1299,·)$, $\chi_{2860}(1301,·)$, $\chi_{2860}(2521,·)$, $\chi_{2860}(1039,·)$, $\chi_{2860}(1119,·)$, $\chi_{2860}(1819,·)$, $\chi_{2860}(2341,·)$, $\chi_{2860}(2599,·)$, $\chi_{2860}(2601,·)$, $\chi_{2860}(1899,·)$, $\chi_{2860}(2419,·)$, $\chi_{2860}(521,·)$, $\chi_{2860}(2679,·)$, $\chi_{2860}(701,·)$, $\chi_{2860}(2261,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{2}$, $\frac{1}{253} a^{14} - \frac{1}{253} a^{13} - \frac{8}{253} a^{12} + \frac{7}{253} a^{11} + \frac{1}{253} a^{3} - \frac{1}{253} a^{2} - \frac{8}{253} a + \frac{7}{253}$, $\frac{1}{8349} a^{15} + \frac{5}{2783} a^{14} - \frac{346}{8349} a^{13} - \frac{29}{8349} a^{12} + \frac{89}{8349} a^{11} - \frac{1}{33} a^{10} + \frac{7}{33} a^{9} + \frac{8}{33} a^{8} + \frac{13}{33} a^{7} - \frac{10}{33} a^{6} + \frac{13}{33} a^{5} + \frac{1181}{2783} a^{4} + \frac{268}{8349} a^{3} + \frac{3955}{8349} a^{2} + \frac{4019}{8349} a + \frac{2872}{8349}$, $\frac{1}{1257668338047} a^{16} + \frac{71370832}{1257668338047} a^{15} + \frac{2053683059}{1257668338047} a^{14} + \frac{11870155084}{419222779349} a^{13} + \frac{2189334398}{419222779349} a^{12} + \frac{32271646375}{1257668338047} a^{11} - \frac{5058205}{1657007033} a^{10} + \frac{726503915}{1657007033} a^{9} - \frac{438857918}{1657007033} a^{8} - \frac{324665296}{1657007033} a^{7} - \frac{382649970}{1657007033} a^{6} - \frac{54314521205}{1257668338047} a^{5} + \frac{616268568217}{1257668338047} a^{4} + \frac{45356218610}{1257668338047} a^{3} - \frac{83219477972}{419222779349} a^{2} - \frac{198444441483}{419222779349} a + \frac{20789211268}{1257668338047}$, $\frac{1}{1257668338047} a^{17} - \frac{36230099}{1257668338047} a^{15} + \frac{2409318520}{1257668338047} a^{14} + \frac{14936537524}{419222779349} a^{13} + \frac{27166753414}{1257668338047} a^{12} + \frac{7299140921}{1257668338047} a^{11} + \frac{66877828}{1657007033} a^{10} + \frac{554868307}{1657007033} a^{9} + \frac{489621417}{1657007033} a^{8} + \frac{373823402}{1657007033} a^{7} - \frac{474653404235}{1257668338047} a^{6} - \frac{720634416}{1657007033} a^{5} + \frac{99470234959}{1257668338047} a^{4} - \frac{579669058118}{1257668338047} a^{3} + \frac{79764624012}{419222779349} a^{2} + \frac{548771098423}{1257668338047} a - \frac{136979652562}{1257668338047}$, $\frac{1}{28926371775081} a^{18} - \frac{1}{9642123925027} a^{16} + \frac{116218595}{28926371775081} a^{15} + \frac{29005566016}{28926371775081} a^{14} + \frac{1030259154713}{28926371775081} a^{13} - \frac{46956989209}{2629670161371} a^{12} + \frac{349206404793}{9642123925027} a^{11} + \frac{1366048933}{114333485277} a^{10} + \frac{39047889464}{114333485277} a^{9} - \frac{37706877254}{114333485277} a^{8} + \frac{1895408419568}{9642123925027} a^{7} + \frac{11091846880}{114333485277} a^{6} + \frac{3771717418202}{28926371775081} a^{5} + \frac{1390551755244}{9642123925027} a^{4} - \frac{2461613651769}{9642123925027} a^{3} - \frac{1765522021990}{9642123925027} a^{2} + \frac{109134147854}{876556720457} a - \frac{4132753690823}{28926371775081}$, $\frac{1}{2399504182745556677555731780868654937864483573161861024686461} a^{19} + \frac{7186370096687750418930560481160469918918843773}{799834727581852225851910593622884979288161191053953674895487} a^{18} + \frac{298351877926460879755614311373485121696703675982}{799834727581852225851910593622884979288161191053953674895487} a^{17} - \frac{6899746814235807724347354276991639742547067723}{104326268815024203371988338298637171211499285789646131508107} a^{16} + \frac{38443644452426104731417222730347533907314041585746609879}{799834727581852225851910593622884979288161191053953674895487} a^{15} + \frac{668218515495502685187967234976083419279554171260654466370}{799834727581852225851910593622884979288161191053953674895487} a^{14} + \frac{73590064606608098451519354715632835135463089767845193978403}{2399504182745556677555731780868654937864483573161861024686461} a^{13} - \frac{32210848591243359680667222990941773006583983091238073203685}{799834727581852225851910593622884979288161191053953674895487} a^{12} - \frac{11932135761577537152892546112170672535828183932019546325124}{799834727581852225851910593622884979288161191053953674895487} a^{11} - \frac{184261090398711434419470210813332617316703681935861607514}{9484206255911291215635303481694288291954480526331466500737} a^{10} - \frac{4223981618734496357098001165519017561770848717851440462749}{9484206255911291215635303481694288291954480526331466500737} a^{9} + \frac{340630191542312582493100165672240236653130904327233012649117}{799834727581852225851910593622884979288161191053953674895487} a^{8} - \frac{336420746192496201786667056110609185005745100774060011072439}{2399504182745556677555731780868654937864483573161861024686461} a^{7} + \frac{715555292959470192796679614518753543721220292859038166036112}{2399504182745556677555731780868654937864483573161861024686461} a^{6} - \frac{15476696677058289517143970199456774951628781036127169791672}{104326268815024203371988338298637171211499285789646131508107} a^{5} + \frac{898646437520616064401012287341241030906032974090230021009642}{2399504182745556677555731780868654937864483573161861024686461} a^{4} + \frac{381604932911597332914003708383060660721971922204121335008218}{2399504182745556677555731780868654937864483573161861024686461} a^{3} + \frac{315468607232463642745030612799994183392863593626965371101386}{799834727581852225851910593622884979288161191053953674895487} a^{2} + \frac{1197529333912252568559863390296286288580197548024091552035837}{2399504182745556677555731780868654937864483573161861024686461} a - \frac{329715104781029618729905835736216343000123298921695216948876}{2399504182745556677555731780868654937864483573161861024686461}$

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

Class group and class number

$C_{10}\times C_{156200}$, which has order $1562000$ (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

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{55}, \sqrt{-65})\), \(\Q(\zeta_{11})^+\), 10.0.875489472034463.1, 10.0.254687846410025600000.1, 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$13$13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$