↑

Properties

Label 20.0.94964489254...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $177.37$
Ramified primes $2, 3, 5, 7, 11$
Class number $15991712$ (GRH)
Class group $[2, 4, 1998964]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1181640625, 0, 5908203125, 0, 8791406250, 0, 5813671875, 0, 1958687500, 0, 347703125, 0, 31501250, 0, 1361250, 0, 28325, 0, 275, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 275*x^18 + 28325*x^16 + 1361250*x^14 + 31501250*x^12 + 347703125*x^10 + 1958687500*x^8 + 5813671875*x^6 + 8791406250*x^4 + 5908203125*x^2 + 1181640625)
 
gp: K = bnfinit(x^20 + 275*x^18 + 28325*x^16 + 1361250*x^14 + 31501250*x^12 + 347703125*x^10 + 1958687500*x^8 + 5813671875*x^6 + 8791406250*x^4 + 5908203125*x^2 + 1181640625, 1)
 

Normalized defining polynomial

\( x^{20} + 275 x^{18} + 28325 x^{16} + 1361250 x^{14} + 31501250 x^{12} + 347703125 x^{10} + 1958687500 x^{8} + 5813671875 x^{6} + 8791406250 x^{4} + 5908203125 x^{2} + 1181640625 \)

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:  \(949644892545940254829738055123773440000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.37$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(3079,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(139,·)$, $\chi_{4620}(3599,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(659,·)$, $\chi_{4620}(1301,·)$, $\chi_{4620}(4439,·)$, $\chi_{4620}(2339,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(3821,·)$, $\chi_{4620}(239,·)$, $\chi_{4620}(881,·)$, $\chi_{4620}(2659,·)$, $\chi_{4620}(2561,·)$, $\chi_{4620}(1399,·)$, $\chi_{4620}(1721,·)$, $\chi_{4620}(2941,·)$, $\chi_{4620}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{34375} a^{10}$, $\frac{1}{34375} a^{11}$, $\frac{1}{171875} a^{12}$, $\frac{1}{171875} a^{13}$, $\frac{1}{859375} a^{14}$, $\frac{1}{859375} a^{15}$, $\frac{1}{184765625} a^{16} + \frac{17}{36953125} a^{14} - \frac{1}{671875} a^{12} + \frac{4}{1478125} a^{10} - \frac{3}{5375} a^{6} - \frac{14}{1075} a^{4} + \frac{19}{215} a^{2} - \frac{3}{43}$, $\frac{1}{184765625} a^{17} + \frac{17}{36953125} a^{15} - \frac{1}{671875} a^{13} + \frac{4}{1478125} a^{11} - \frac{3}{5375} a^{7} - \frac{14}{1075} a^{5} + \frac{19}{215} a^{3} - \frac{3}{43} a$, $\frac{1}{1402889545169921875} a^{18} - \frac{4994864}{2244623272271875} a^{16} - \frac{16772244999}{56115581806796875} a^{14} + \frac{2190436046}{2244623272271875} a^{12} + \frac{355784383}{89784930890875} a^{10} - \frac{17531407367}{40811332223125} a^{8} + \frac{4466090917}{1632453288925} a^{6} - \frac{3275158719}{326490657785} a^{4} + \frac{18279737144}{326490657785} a^{2} - \frac{16500676438}{65298131557}$, $\frac{1}{1402889545169921875} a^{19} - \frac{4994864}{2244623272271875} a^{17} - \frac{16772244999}{56115581806796875} a^{15} + \frac{2190436046}{2244623272271875} a^{13} + \frac{355784383}{89784930890875} a^{11} - \frac{17531407367}{40811332223125} a^{9} + \frac{4466090917}{1632453288925} a^{7} - \frac{3275158719}{326490657785} a^{5} + \frac{18279737144}{326490657785} a^{3} - \frac{16500676438}{65298131557} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{1998964}$, which has order $15991712$ (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:  \( 5868059.799558259 \) (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

\(\Q(\sqrt{-385}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{21}, \sqrt{-165})\), \(\Q(\zeta_{11})^+\), 10.0.126816085896438400000.1, 10.0.1833540124521600000.1, 10.10.875463320250981.1

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 R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
11Data not computed