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Properties

Label 20.20.2470566082...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 331^{2}\cdot 39161^{2}$
Root discriminant $234.23$
Ramified primes $2, 5, 11, 331, 39161$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times C_2^4:C_5$ (as 20T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![525065587402128125, 0, -140883697770453125, 0, 15118770876116875, 0, -864153686645000, 0, 29331554339500, 0, -620327617375, 0, 8294073900, 0, -69250575, 0, 345290, 0, -925, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 925*x^18 + 345290*x^16 - 69250575*x^14 + 8294073900*x^12 - 620327617375*x^10 + 29331554339500*x^8 - 864153686645000*x^6 + 15118770876116875*x^4 - 140883697770453125*x^2 + 525065587402128125)
 
gp: K = bnfinit(x^20 - 925*x^18 + 345290*x^16 - 69250575*x^14 + 8294073900*x^12 - 620327617375*x^10 + 29331554339500*x^8 - 864153686645000*x^6 + 15118770876116875*x^4 - 140883697770453125*x^2 + 525065587402128125, 1)
 

Normalized defining polynomial

\( x^{20} - 925 x^{18} + 345290 x^{16} - 69250575 x^{14} + 8294073900 x^{12} - 620327617375 x^{10} + 29331554339500 x^{8} - 864153686645000 x^{6} + 15118770876116875 x^{4} - 140883697770453125 x^{2} + 525065587402128125 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(247056608274284803549371247668512000000000000000=2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 331^{2}\cdot 39161^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $234.23$
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:  $2, 5, 11, 331, 39161$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{2183935835235625} a^{16} - \frac{1306222718786}{436787167047125} a^{14} - \frac{15705465842}{17471486681885} a^{12} + \frac{1377389274733}{87357433409425} a^{10} + \frac{510581108656}{87357433409425} a^{8} - \frac{277821184323}{17471486681885} a^{6} + \frac{865009508558}{17471486681885} a^{4} - \frac{155051145140}{3494297336377} a^{2} - \frac{225205033322}{3494297336377}$, $\frac{1}{2183935835235625} a^{17} - \frac{1306222718786}{436787167047125} a^{15} - \frac{15705465842}{17471486681885} a^{13} + \frac{1377389274733}{87357433409425} a^{11} + \frac{510581108656}{87357433409425} a^{9} - \frac{277821184323}{17471486681885} a^{7} + \frac{865009508558}{17471486681885} a^{5} - \frac{155051145140}{3494297336377} a^{3} - \frac{225205033322}{3494297336377} a$, $\frac{1}{181483627436769403907846022557446031098969094076875} a^{18} - \frac{4902207004859019397016285695768311}{36296725487353880781569204511489206219793818815375} a^{16} + \frac{53556534000464597879947972547429061649777361929}{36296725487353880781569204511489206219793818815375} a^{14} - \frac{108935094823705830838347009397232342463491745352}{36296725487353880781569204511489206219793818815375} a^{12} + \frac{52388367822222432382307166510139316181056729344}{7259345097470776156313840902297841243958763763075} a^{10} + \frac{5823401147344957590012579184424599198661346568}{7259345097470776156313840902297841243958763763075} a^{8} + \frac{4815571160529660404448203563882217910269599757}{290373803898831046252553636091913649758350550523} a^{6} - \frac{12187262468363361509617346537923606077789441068}{290373803898831046252553636091913649758350550523} a^{4} + \frac{94587474155000750568390936679591380459184796452}{290373803898831046252553636091913649758350550523} a^{2} + \frac{4061787851690432430475122070407630598965}{22401426098120389848719924285908536520153}$, $\frac{1}{181483627436769403907846022557446031098969094076875} a^{19} - \frac{4902207004859019397016285695768311}{36296725487353880781569204511489206219793818815375} a^{17} + \frac{53556534000464597879947972547429061649777361929}{36296725487353880781569204511489206219793818815375} a^{15} - \frac{108935094823705830838347009397232342463491745352}{36296725487353880781569204511489206219793818815375} a^{13} + \frac{52388367822222432382307166510139316181056729344}{7259345097470776156313840902297841243958763763075} a^{11} + \frac{5823401147344957590012579184424599198661346568}{7259345097470776156313840902297841243958763763075} a^{9} + \frac{4815571160529660404448203563882217910269599757}{290373803898831046252553636091913649758350550523} a^{7} - \frac{12187262468363361509617346537923606077789441068}{290373803898831046252553636091913649758350550523} a^{5} + \frac{94587474155000750568390936679591380459184796452}{290373803898831046252553636091913649758350550523} a^{3} + \frac{4061787851690432430475122070407630598965}{22401426098120389848719924285908536520153} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 76575733458300000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_2^4:C_5$ (as 20T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_4\times C_2^4:C_5$
Character table for $C_4\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
331Data not computed
39161Data not computed