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Properties

Label 20.0.16024335156...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{23}\cdot 29^{5}$
Root discriminant $25.72$
Ramified primes $2, 5, 29$
Class number $2$
Class group $[2]$
Galois group $C_2^2:F_5$ (as 20T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431, 445, 1760, 2385, 5420, 3749, 5100, 2575, 2595, -1430, -384, -180, 895, -235, -50, -51, 70, -25, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 - 25*x^17 + 70*x^16 - 51*x^15 - 50*x^14 - 235*x^13 + 895*x^12 - 180*x^11 - 384*x^10 - 1430*x^9 + 2595*x^8 + 2575*x^7 + 5100*x^6 + 3749*x^5 + 5420*x^4 + 2385*x^3 + 1760*x^2 + 445*x + 431)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 - 25*x^17 + 70*x^16 - 51*x^15 - 50*x^14 - 235*x^13 + 895*x^12 - 180*x^11 - 384*x^10 - 1430*x^9 + 2595*x^8 + 2575*x^7 + 5100*x^6 + 3749*x^5 + 5420*x^4 + 2385*x^3 + 1760*x^2 + 445*x + 431, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} - 25 x^{17} + 70 x^{16} - 51 x^{15} - 50 x^{14} - 235 x^{13} + 895 x^{12} - 180 x^{11} - 384 x^{10} - 1430 x^{9} + 2595 x^{8} + 2575 x^{7} + 5100 x^{6} + 3749 x^{5} + 5420 x^{4} + 2385 x^{3} + 1760 x^{2} + 445 x + 431 \)

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:  \(16024335156250000000000000000=2^{16}\cdot 5^{23}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.72$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{3382} a^{18} + \frac{364}{1691} a^{17} + \frac{139}{3382} a^{16} - \frac{151}{3382} a^{15} - \frac{540}{1691} a^{14} - \frac{107}{3382} a^{13} + \frac{583}{1691} a^{12} + \frac{683}{1691} a^{11} - \frac{559}{1691} a^{10} + \frac{671}{1691} a^{9} + \frac{206}{1691} a^{8} - \frac{807}{1691} a^{7} - \frac{527}{3382} a^{6} - \frac{601}{1691} a^{5} - \frac{587}{3382} a^{4} + \frac{1271}{3382} a^{3} - \frac{620}{1691} a^{2} + \frac{5}{178} a - \frac{123}{1691}$, $\frac{1}{209958259999223267288606372435623554} a^{19} + \frac{1844961814773452244744460119556}{104979129999611633644303186217811777} a^{18} + \frac{2628821632523793823266024727111272}{34993043333203877881434395405937259} a^{17} + \frac{8091321026461770286210801249918411}{104979129999611633644303186217811777} a^{16} + \frac{5770179181765649767192261828879347}{69986086666407755762868790811874518} a^{15} - \frac{12937850378687452546567231655912978}{34993043333203877881434395405937259} a^{14} + \frac{21684683858574140265110500648451425}{209958259999223267288606372435623554} a^{13} + \frac{28520744018311975704376916951492519}{69986086666407755762868790811874518} a^{12} + \frac{22034465522341097448720541017017078}{104979129999611633644303186217811777} a^{11} + \frac{46718746872806032223840973276227855}{104979129999611633644303186217811777} a^{10} + \frac{52396936123504136560206321043528823}{104979129999611633644303186217811777} a^{9} + \frac{43022917063661505246813881501227693}{104979129999611633644303186217811777} a^{8} - \frac{69186519659554886803402747081024399}{209958259999223267288606372435623554} a^{7} + \frac{15429192155260505701865147608197786}{34993043333203877881434395405937259} a^{6} + \frac{3814275189386882373337550382957430}{34993043333203877881434395405937259} a^{5} - \frac{28148943971703974892369205580292827}{104979129999611633644303186217811777} a^{4} + \frac{86678321330749386989086576840116739}{209958259999223267288606372435623554} a^{3} + \frac{15874225434686953051143406998291281}{104979129999611633644303186217811777} a^{2} - \frac{1191612551708676167692169715135}{44323044120587559064514750355842} a - \frac{95260780773334977752660704519396505}{209958259999223267288606372435623554}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 615825.309667 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3625.1, 5.1.50000.1, 10.2.12500000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$