↑

Properties

Label 20.0.62504128134...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{31}\cdot 41^{10}$
Root discriminant $77.59$
Ramified primes $5, 41$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1114173964016, 1241779064960, 751545978640, 280542235970, 49933574625, -10776463559, -10452282415, -4046349655, -744886310, -46253755, 39844916, 12199900, 3060265, 229210, 43460, -4369, 1680, -125, 55, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 55*x^18 - 125*x^17 + 1680*x^16 - 4369*x^15 + 43460*x^14 + 229210*x^13 + 3060265*x^12 + 12199900*x^11 + 39844916*x^10 - 46253755*x^9 - 744886310*x^8 - 4046349655*x^7 - 10452282415*x^6 - 10776463559*x^5 + 49933574625*x^4 + 280542235970*x^3 + 751545978640*x^2 + 1241779064960*x + 1114173964016)
 
gp: K = bnfinit(x^20 - 5*x^19 + 55*x^18 - 125*x^17 + 1680*x^16 - 4369*x^15 + 43460*x^14 + 229210*x^13 + 3060265*x^12 + 12199900*x^11 + 39844916*x^10 - 46253755*x^9 - 744886310*x^8 - 4046349655*x^7 - 10452282415*x^6 - 10776463559*x^5 + 49933574625*x^4 + 280542235970*x^3 + 751545978640*x^2 + 1241779064960*x + 1114173964016, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 55 x^{18} - 125 x^{17} + 1680 x^{16} - 4369 x^{15} + 43460 x^{14} + 229210 x^{13} + 3060265 x^{12} + 12199900 x^{11} + 39844916 x^{10} - 46253755 x^{9} - 744886310 x^{8} - 4046349655 x^{7} - 10452282415 x^{6} - 10776463559 x^{5} + 49933574625 x^{4} + 280542235970 x^{3} + 751545978640 x^{2} + 1241779064960 x + 1114173964016 \)

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:  \(62504128134587784297764301300048828125=5^{31}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.59$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{76} a^{17} + \frac{9}{76} a^{16} + \frac{9}{76} a^{15} - \frac{1}{76} a^{14} - \frac{9}{76} a^{13} + \frac{15}{76} a^{12} + \frac{5}{38} a^{11} + \frac{9}{38} a^{10} + \frac{4}{19} a^{9} + \frac{5}{38} a^{8} - \frac{1}{4} a^{7} - \frac{15}{76} a^{6} - \frac{1}{4} a^{5} + \frac{9}{76} a^{4} + \frac{1}{4} a^{3} - \frac{1}{76} a^{2} - \frac{2}{19} a + \frac{8}{19}$, $\frac{1}{988} a^{18} + \frac{1}{247} a^{17} + \frac{10}{247} a^{16} - \frac{2}{247} a^{15} - \frac{21}{494} a^{14} - \frac{61}{247} a^{13} - \frac{27}{988} a^{12} + \frac{49}{247} a^{11} - \frac{113}{494} a^{10} + \frac{3}{494} a^{9} + \frac{121}{988} a^{8} + \frac{59}{494} a^{7} + \frac{85}{494} a^{6} + \frac{2}{19} a^{5} + \frac{25}{494} a^{4} + \frac{14}{247} a^{3} + \frac{111}{988} a^{2} - \frac{20}{247} a - \frac{116}{247}$, $\frac{1}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{19} - \frac{336742486643361141727370861209131759275970663803000146004662146408966037513405464344454866938248137}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{18} + \frac{7166953274401115387519686521936547781761982520207751429011643243976061892557769653286427829142132133}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{17} + \frac{71243382136586228486313355584003341641184484180471941958140032925264228663913469083985224122781548991}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{16} - \frac{573488648445213621393116303815707806383772203134372312842081675498510253860114703829309170798447847}{8840902688866713415158585686147776362714681875617253171754353165474991317199329284874824694525258233} a^{15} - \frac{10502836703383129130516395158872380896933408056119159630836493176331682829320225859997947004938291677}{103370554515980033777238848022650923625587049622601729392820129319399898478022927023151796428295327032} a^{14} - \frac{50656923265628212441090914148283377951051297810601876135355333335732667978516913485617900313303916027}{671908604353870219552052512147231003566315822546911241053330840576099340107149025650486676783919625708} a^{13} + \frac{10331591488457093499886711630499014392778792869780028538459993875713248917351690726269923203972671883}{51685277257990016888619424011325461812793524811300864696410064659699949239011463511575898214147663516} a^{12} + \frac{222831201727215908718817380693731896222700822648686206950998383443473326623037729033499593036389183527}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{11} - \frac{76327752944345881679594268595857635764858281259503091129504353262428173321680421513462980551981125849}{335954302176935109776026256073615501783157911273455620526665420288049670053574512825243338391959812854} a^{10} - \frac{119834531801705071299063016150322447756857342091263683259015007006692939199644905506256160570189206493}{671908604353870219552052512147231003566315822546911241053330840576099340107149025650486676783919625708} a^{9} + \frac{205747545223058096405789425062719515063389392364231559265997599047210552537427936689492875497877152777}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{8} - \frac{106760745025919743440237905484037717984858923860021060496523969756127726698472675387448627747835004303}{671908604353870219552052512147231003566315822546911241053330840576099340107149025650486676783919625708} a^{7} + \frac{297648932060418262726948308975940598393884759995892333065968199920943365648006301874323148192785381769}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{6} + \frac{459792812004533020229088438789021880176834903569326862312142385365149029470416207817200375783912137087}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{5} + \frac{380571778700353904337180898632108817602097163981321477877759520537438283770714049429832574827867813885}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{4} - \frac{53770384127629100244923878457871720876640710269065069778592912950967803043269213567635421564559723999}{1343817208707740439104105024294462007132631645093822482106661681152198680214298051300973353567839251416} a^{3} - \frac{194128201057853881734118838937489227862610794170784157941341306994619081755414803245718749223255517293}{671908604353870219552052512147231003566315822546911241053330840576099340107149025650486676783919625708} a^{2} + \frac{24945120133845945290827656960204295900233806357691160015152940876832537249282649878658557425256401891}{335954302176935109776026256073615501783157911273455620526665420288049670053574512825243338391959812854} a + \frac{48260982737669313756822095382025503784270529697910374770137566463034148165245936703156705279774706091}{167977151088467554888013128036807750891578955636727810263332710144024835026787256412621669195979906427}$

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

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.210125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$41$41.10.5.1$x^{10} - 3362 x^{6} + 2825761 x^{2} - 5676953849$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41.10.5.1$x^{10} - 3362 x^{6} + 2825761 x^{2} - 5676953849$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$