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Properties

Label 20.20.8088551234...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{12}\cdot 257^{4}\cdot 431^{6}\cdot 1031^{4}$
Root discriminant $393.91$
Ramified primes $2, 5, 257, 431, 1031$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![102561320445857296, 0, -28079433439933088, 0, 3145361149528271, 0, -193051166481831, 0, 7257234585990, 0, -175645980387, 0, 2783036051, 0, -28582758, 0, 182428, 0, -654, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 654*x^18 + 182428*x^16 - 28582758*x^14 + 2783036051*x^12 - 175645980387*x^10 + 7257234585990*x^8 - 193051166481831*x^6 + 3145361149528271*x^4 - 28079433439933088*x^2 + 102561320445857296)
 
gp: K = bnfinit(x^20 - 654*x^18 + 182428*x^16 - 28582758*x^14 + 2783036051*x^12 - 175645980387*x^10 + 7257234585990*x^8 - 193051166481831*x^6 + 3145361149528271*x^4 - 28079433439933088*x^2 + 102561320445857296, 1)
 

Normalized defining polynomial

\( x^{20} - 654 x^{18} + 182428 x^{16} - 28582758 x^{14} + 2783036051 x^{12} - 175645980387 x^{10} + 7257234585990 x^{8} - 193051166481831 x^{6} + 3145361149528271 x^{4} - 28079433439933088 x^{2} + 102561320445857296 \)

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:  \(8088551234301552526733640713612755353856000000000000=2^{20}\cdot 5^{12}\cdot 257^{4}\cdot 431^{6}\cdot 1031^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $393.91$
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:  $2, 5, 257, 431, 1031$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{431} a^{10} + \frac{208}{431} a^{8} + \frac{115}{431} a^{6} - \frac{131}{431} a^{4} + \frac{91}{431} a^{2}$, $\frac{1}{431} a^{11} + \frac{208}{431} a^{9} + \frac{115}{431} a^{7} - \frac{131}{431} a^{5} + \frac{91}{431} a^{3}$, $\frac{1}{185761} a^{12} + \frac{208}{185761} a^{10} - \frac{9798}{185761} a^{8} - \frac{62195}{185761} a^{6} + \frac{37588}{185761} a^{4} + \frac{93}{431} a^{2}$, $\frac{1}{185761} a^{13} + \frac{208}{185761} a^{11} - \frac{9798}{185761} a^{9} - \frac{62195}{185761} a^{7} + \frac{37588}{185761} a^{5} + \frac{93}{431} a^{3}$, $\frac{1}{240188973} a^{14} + \frac{213}{80062991} a^{12} - \frac{105911}{240188973} a^{10} - \frac{77103445}{240188973} a^{8} - \frac{3548332}{240188973} a^{6} - \frac{246779}{557283} a^{4} + \frac{463}{1293} a^{2} - \frac{1}{3}$, $\frac{1}{240188973} a^{15} + \frac{213}{80062991} a^{13} - \frac{105911}{240188973} a^{11} - \frac{77103445}{240188973} a^{9} - \frac{3548332}{240188973} a^{7} - \frac{246779}{557283} a^{5} + \frac{463}{1293} a^{3} - \frac{1}{3} a$, $\frac{1}{310564342089} a^{16} + \frac{208}{310564342089} a^{14} + \frac{175963}{310564342089} a^{12} + \frac{9384340}{34507149121} a^{10} + \frac{42899682928}{310564342089} a^{8} - \frac{8570773}{720566919} a^{6} + \frac{51473}{557283} a^{4} - \frac{1802}{3879} a^{2} - \frac{2}{9}$, $\frac{1}{621128684178} a^{17} + \frac{104}{310564342089} a^{15} - \frac{747943}{310564342089} a^{13} - \frac{14626974}{34507149121} a^{11} - \frac{251283882659}{621128684178} a^{9} - \frac{487883287}{1441133838} a^{7} + \frac{247996}{557283} a^{5} - \frac{2639}{7758} a^{3} + \frac{7}{18} a$, $\frac{1}{480435048672459474216341591494613496} a^{18} - \frac{184204589336149146853757}{240217524336229737108170795747306748} a^{16} + \frac{72668708384219717096743457}{120108762168114868554085397873653374} a^{14} + \frac{264350970498411311197502556977}{240217524336229737108170795747306748} a^{12} - \frac{140332984405624027703687539122941}{480435048672459474216341591494613496} a^{10} - \frac{354265718698514631109545131636945}{1114698488799209916975270513908616} a^{8} + \frac{600013711446917040506739426113}{1293153699303027745910986675068} a^{6} + \frac{1428214673351007588491310943}{6000713221823794644598546056} a^{4} + \frac{101356220443278416055097}{4640922832036964148954792} a^{2} - \frac{1198583711032608350029}{8075851795887988658274}$, $\frac{1}{960870097344918948432683182989226992} a^{19} - \frac{184204589336149146853757}{480435048672459474216341591494613496} a^{17} + \frac{72668708384219717096743457}{240217524336229737108170795747306748} a^{15} - \frac{1028802728804616434713484118091}{480435048672459474216341591494613496} a^{13} + \frac{436413565483526346972612517957387}{960870097344918948432683182989226992} a^{11} - \frac{872217280440235465858068634460585}{2229396977598419833950541027817232} a^{9} + \frac{84864315706565964526534081925}{2586307398606055491821973350136} a^{7} + \frac{4390824752968924451801811295}{12001426443647589289197092112} a^{5} - \frac{4561102216382720369321759}{9281845664073928297909584} a^{3} + \frac{6877268084855380308245}{16151703591775977316548} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 102519291655000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.1096992360765625.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
257Data not computed
431Data not computed
1031Data not computed