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Properties

Label 20.0.90565194372...625.11
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $88.68$
Ramified primes $3, 5, 7, 11$
Class number $634040$ (GRH)
Class group $[22, 28820]$ (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![21859650901, -9593112870, 16343976550, -6654062822, 5878132623, -2170254966, 1325598923, -437116790, 206034873, -59959174, 22954306, -5814310, 1850084, -399434, 106164, -18820, 4129, -556, 97, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 97*x^18 - 556*x^17 + 4129*x^16 - 18820*x^15 + 106164*x^14 - 399434*x^13 + 1850084*x^12 - 5814310*x^11 + 22954306*x^10 - 59959174*x^9 + 206034873*x^8 - 437116790*x^7 + 1325598923*x^6 - 2170254966*x^5 + 5878132623*x^4 - 6654062822*x^3 + 16343976550*x^2 - 9593112870*x + 21859650901)
 
gp: K = bnfinit(x^20 - 8*x^19 + 97*x^18 - 556*x^17 + 4129*x^16 - 18820*x^15 + 106164*x^14 - 399434*x^13 + 1850084*x^12 - 5814310*x^11 + 22954306*x^10 - 59959174*x^9 + 206034873*x^8 - 437116790*x^7 + 1325598923*x^6 - 2170254966*x^5 + 5878132623*x^4 - 6654062822*x^3 + 16343976550*x^2 - 9593112870*x + 21859650901, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 97 x^{18} - 556 x^{17} + 4129 x^{16} - 18820 x^{15} + 106164 x^{14} - 399434 x^{13} + 1850084 x^{12} - 5814310 x^{11} + 22954306 x^{10} - 59959174 x^{9} + 206034873 x^{8} - 437116790 x^{7} + 1325598923 x^{6} - 2170254966 x^{5} + 5878132623 x^{4} - 6654062822 x^{3} + 16343976550 x^{2} - 9593112870 x + 21859650901 \)

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:  \(905651943727436308698404364703916015625=3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.68$
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:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1155=3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1155}(1,·)$, $\chi_{1155}(1154,·)$, $\chi_{1155}(454,·)$, $\chi_{1155}(841,·)$, $\chi_{1155}(524,·)$, $\chi_{1155}(526,·)$, $\chi_{1155}(596,·)$, $\chi_{1155}(664,·)$, $\chi_{1155}(281,·)$, $\chi_{1155}(734,·)$, $\chi_{1155}(1121,·)$, $\chi_{1155}(34,·)$, $\chi_{1155}(421,·)$, $\chi_{1155}(874,·)$, $\chi_{1155}(491,·)$, $\chi_{1155}(559,·)$, $\chi_{1155}(629,·)$, $\chi_{1155}(631,·)$, $\chi_{1155}(314,·)$, $\chi_{1155}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{19} + \frac{28673975021521432647520091209456077986551711712407819726337404953205}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{18} - \frac{22939133514464943284131422689007635583656299310388653544123053565849}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{17} - \frac{55792356439734263048152931473413505314659581762637950159868829475027}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{16} - \frac{110714234472212707745090256749691759114616079423706419623153907066302}{599360114335753652601935389546034252405922753567833508155757055268717} a^{15} + \frac{160853617223355050636036402294890821744118591844790338207564139041099}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{14} - \frac{130324203173367628865032956585598416150015749737157050920734548060601}{599360114335753652601935389546034252405922753567833508155757055268717} a^{13} - \frac{6461440231423780925000842010877396266341840629116015269192139834652}{599360114335753652601935389546034252405922753567833508155757055268717} a^{12} - \frac{190512838253445581997664373687370755468499142802924167206772661804577}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{11} + \frac{3836095368194631760111194098861383199702151952667339174154284446771}{26059135405902332721823277806349315321996641459471022093728567620379} a^{10} - \frac{259880610760414222409840015888385257625148666029332416699999557900157}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{9} - \frac{443068377887953512923380157406941869585660669135334407364969070534623}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{8} - \frac{439848283029275208633936484941265345405249888420212757840739363440923}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{7} - \frac{285126800582767000212460081388483816272453621316944589310475674667036}{599360114335753652601935389546034252405922753567833508155757055268717} a^{6} + \frac{351102869736243795316510386168389537887300118140776932687575993252085}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{5} + \frac{280496341956265020384041513374603591228704495837906545084968915794507}{599360114335753652601935389546034252405922753567833508155757055268717} a^{4} - \frac{49379515219821617887157766930666127117664315215818049823847652523197}{1198720228671507305203870779092068504811845507135667016311514110537434} a^{3} + \frac{8451567031893140096400015711732129372939164842312518905887144914189}{52118270811804665443646555612698630643993282918942044187457135240758} a^{2} - \frac{108493819995256308204519196814574711584274861920567992588327997527487}{599360114335753652601935389546034252405922753567833508155757055268717} a - \frac{208074062598567938885759540390250106276729473328631073281541122202374}{599360114335753652601935389546034252405922753567833508155757055268717}$

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

Class group and class number

$C_{22}\times C_{28820}$, which has order $634040$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-1155}) \), \(\Q(\sqrt{33}, \sqrt{-35})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\), 10.0.11258530353021875.4, 10.0.30094051633627471875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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
7Data 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}$