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Properties

Label 20.0.11753236087...4144.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}\cdot 17^{10}$
Root discriminant $71.37$
Ramified primes $2, 11, 17$
Class number $45100$ (GRH)
Class group $[5, 9020]$ (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![57592921, 0, 81835911, 0, 52211443, 0, 19589641, 0, 4742038, 0, 760574, 0, 79368, 0, 5124, 0, 225, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15*x^18 + 225*x^16 + 5124*x^14 + 79368*x^12 + 760574*x^10 + 4742038*x^8 + 19589641*x^6 + 52211443*x^4 + 81835911*x^2 + 57592921)
 
gp: K = bnfinit(x^20 + 15*x^18 + 225*x^16 + 5124*x^14 + 79368*x^12 + 760574*x^10 + 4742038*x^8 + 19589641*x^6 + 52211443*x^4 + 81835911*x^2 + 57592921, 1)
 

Normalized defining polynomial

\( x^{20} + 15 x^{18} + 225 x^{16} + 5124 x^{14} + 79368 x^{12} + 760574 x^{10} + 4742038 x^{8} + 19589641 x^{6} + 52211443 x^{4} + 81835911 x^{2} + 57592921 \)

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:  \(11753236087178400090650813750280454144=2^{20}\cdot 11^{18}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.37$
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:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(748=2^{2}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{748}(1,·)$, $\chi_{748}(67,·)$, $\chi_{748}(69,·)$, $\chi_{748}(135,·)$, $\chi_{748}(137,·)$, $\chi_{748}(339,·)$, $\chi_{748}(203,·)$, $\chi_{748}(273,·)$, $\chi_{748}(579,·)$, $\chi_{748}(477,·)$, $\chi_{748}(543,·)$, $\chi_{748}(35,·)$, $\chi_{748}(101,·)$, $\chi_{748}(171,·)$, $\chi_{748}(237,·)$, $\chi_{748}(239,·)$, $\chi_{748}(645,·)$, $\chi_{748}(305,·)$, $\chi_{748}(307,·)$, $\chi_{748}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{533723356495195186001142913} a^{18} - \frac{95714155658541307217902952}{533723356495195186001142913} a^{16} + \frac{56873173138044685022512188}{533723356495195186001142913} a^{14} + \frac{196696569135555438962502513}{533723356495195186001142913} a^{12} + \frac{79560431147580873600540571}{533723356495195186001142913} a^{10} + \frac{83093352088794854230870049}{533723356495195186001142913} a^{8} - \frac{248037622067946592027609447}{533723356495195186001142913} a^{6} + \frac{239958713920775336417716215}{533723356495195186001142913} a^{4} - \frac{86533044969803554827305117}{533723356495195186001142913} a^{2} + \frac{201141465901434909381401158}{533723356495195186001142913}$, $\frac{1}{4050426552442036266562673566757} a^{19} - \frac{795343515333499368448920843322}{4050426552442036266562673566757} a^{17} + \frac{1384001536565179161985986085597}{4050426552442036266562673566757} a^{15} + \frac{7135100203573092856977360382}{4050426552442036266562673566757} a^{13} - \frac{980370245450525975810498990610}{4050426552442036266562673566757} a^{11} - \frac{1139949996121648122444210392119}{4050426552442036266562673566757} a^{9} + \frac{1752499465108153044235725716845}{4050426552442036266562673566757} a^{7} - \frac{247941402056344986154113738330}{4050426552442036266562673566757} a^{5} - \frac{1559092457367434941864165753990}{4050426552442036266562673566757} a^{3} - \frac{574618913479423780413849516143}{4050426552442036266562673566757} a$

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

Class group and class number

$C_{5}\times C_{9020}$, which has order $45100$ (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:  \( 281202.490766 \) (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

\(\Q(\sqrt{-187}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{11}, \sqrt{-17})\), \(\Q(\zeta_{11})^+\), 10.0.3347948534700187.1, \(\Q(\zeta_{44})^+\), 10.0.311663572684817408.3

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
17Data not computed