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Properties

Label 20.0.19031596686...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 19^{5}\cdot 349^{5}\cdot 10771^{5}$
Root discriminant $411.12$
Ramified primes $2, 5, 19, 349, 10771$
Class number Not computed
Class group Not computed
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![364338580053661358017501, 0, 24788321856472354874337, 0, 551837397237621960474, 0, 4647879416771177895, 0, 8237200478417331, 0, -70140281045427, 0, -205994049190, 0, 491864356, 0, 895458, 0, -2047, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2047*x^18 + 895458*x^16 + 491864356*x^14 - 205994049190*x^12 - 70140281045427*x^10 + 8237200478417331*x^8 + 4647879416771177895*x^6 + 551837397237621960474*x^4 + 24788321856472354874337*x^2 + 364338580053661358017501)
 
gp: K = bnfinit(x^20 - 2047*x^18 + 895458*x^16 + 491864356*x^14 - 205994049190*x^12 - 70140281045427*x^10 + 8237200478417331*x^8 + 4647879416771177895*x^6 + 551837397237621960474*x^4 + 24788321856472354874337*x^2 + 364338580053661358017501, 1)
 

Normalized defining polynomial

\( x^{20} - 2047 x^{18} + 895458 x^{16} + 491864356 x^{14} - 205994049190 x^{12} - 70140281045427 x^{10} + 8237200478417331 x^{8} + 4647879416771177895 x^{6} + 551837397237621960474 x^{4} + 24788321856472354874337 x^{2} + 364338580053661358017501 \)

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:  \(19031596686524690994504794774570696832010240000000000=2^{20}\cdot 5^{10}\cdot 19^{5}\cdot 349^{5}\cdot 10771^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $411.12$
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:  $2, 5, 19, 349, 10771$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} + \frac{1}{19} a^{12} + \frac{3}{19} a^{10} + \frac{1}{19} a^{8} + \frac{3}{19} a^{6} + \frac{7}{19} a^{4} + \frac{4}{19} a^{2}$, $\frac{1}{19} a^{15} + \frac{1}{19} a^{13} + \frac{3}{19} a^{11} + \frac{1}{19} a^{9} + \frac{3}{19} a^{7} + \frac{7}{19} a^{5} + \frac{4}{19} a^{3}$, $\frac{1}{71422501} a^{16} - \frac{2047}{71422501} a^{14} + \frac{895458}{71422501} a^{12} - \frac{8093151}{71422501} a^{10} - \frac{11556306}{71422501} a^{8} - \frac{1484520}{3759079} a^{6} + \frac{13206723}{71422501} a^{4} + \frac{24010525}{71422501} a^{2}$, $\frac{1}{71422501} a^{17} - \frac{2047}{71422501} a^{15} + \frac{895458}{71422501} a^{13} - \frac{8093151}{71422501} a^{11} - \frac{11556306}{71422501} a^{9} - \frac{1484520}{3759079} a^{7} + \frac{13206723}{71422501} a^{5} + \frac{24010525}{71422501} a^{3}$, $\frac{1}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{18} + \frac{2280457006980628796055958901655542764184055745382460133759879173900226614119974599606933090}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{16} + \frac{15446174839658103709806715494912205459292457920227960489444060838538606199649804087164882266300516}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{14} + \frac{293207975874645812005101429968126293703135546545578139297855785752095060735365327715722134439315647}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{12} + \frac{218490597081071648144246612212348282414681884372585356980934549414599662505284203451377261502372350}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{10} + \frac{13910574179368262560092132934402636190340839891107466830825495074714343161095133515007176530197668}{44233121347217816428313355403323516570773417286154828454323657361508995734542511807013675381641601} a^{8} + \frac{265055769897585159492536891887833368649935916904092083906597540903554635769644784626178738497951200}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{6} - \frac{40541553489841051805194898127578190333948507532857682296797586730261323535865056469968308846895395}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{4} + \frac{1515178110343726295428589507181142995306839097432392829645281790316935278353880205147578951}{11767010309498102175642851720680389151378148021404931488357562413960705729925471586793912919} a^{2} + \frac{60900175874154577657137485518752129050106952316416530940489088421978485451515875738}{164752145958849889275689907872035860958973531624227657448701809167403781196705385419}$, $\frac{1}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{19} + \frac{2280457006980628796055958901655542764184055745382460133759879173900226614119974599606933090}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{17} + \frac{15446174839658103709806715494912205459292457920227960489444060838538606199649804087164882266300516}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{15} + \frac{293207975874645812005101429968126293703135546545578139297855785752095060735365327715722134439315647}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{13} + \frac{218490597081071648144246612212348282414681884372585356980934549414599662505284203451377261502372350}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{11} + \frac{13910574179368262560092132934402636190340839891107466830825495074714343161095133515007176530197668}{44233121347217816428313355403323516570773417286154828454323657361508995734542511807013675381641601} a^{9} + \frac{265055769897585159492536891887833368649935916904092083906597540903554635769644784626178738497951200}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{7} - \frac{40541553489841051805194898127578190333948507532857682296797586730261323535865056469968308846895395}{840429305597138512137953752663146814844694928436941740632149489868670918956307724333259832251190419} a^{5} + \frac{1515178110343726295428589507181142995306839097432392829645281790316935278353880205147578951}{11767010309498102175642851720680389151378148021404931488357562413960705729925471586793912919} a^{3} + \frac{60900175874154577657137485518752129050106952316416530940489088421978485451515875738}{164752145958849889275689907872035860958973531624227657448701809167403781196705385419} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.223195315625.1

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 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
349Data not computed
10771Data not computed