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Properties

Label 20.0.77457396436...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{13}\cdot 6029^{10}$
Root discriminant $221.03$
Ramified primes $5, 6029$
Class number $16350$ (GRH)
Class group $[16350]$ (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5105600303, -5223386330, 7205970510, -1661558095, 3651466705, 2898777, 1055587375, -159474145, 296502290, -87675235, 54147603, -14618695, 5365250, -1149425, 299040, -48809, 9450, -1090, 155, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 155*x^18 - 1090*x^17 + 9450*x^16 - 48809*x^15 + 299040*x^14 - 1149425*x^13 + 5365250*x^12 - 14618695*x^11 + 54147603*x^10 - 87675235*x^9 + 296502290*x^8 - 159474145*x^7 + 1055587375*x^6 + 2898777*x^5 + 3651466705*x^4 - 1661558095*x^3 + 7205970510*x^2 - 5223386330*x + 5105600303)
 
gp: K = bnfinit(x^20 - 10*x^19 + 155*x^18 - 1090*x^17 + 9450*x^16 - 48809*x^15 + 299040*x^14 - 1149425*x^13 + 5365250*x^12 - 14618695*x^11 + 54147603*x^10 - 87675235*x^9 + 296502290*x^8 - 159474145*x^7 + 1055587375*x^6 + 2898777*x^5 + 3651466705*x^4 - 1661558095*x^3 + 7205970510*x^2 - 5223386330*x + 5105600303, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 155 x^{18} - 1090 x^{17} + 9450 x^{16} - 48809 x^{15} + 299040 x^{14} - 1149425 x^{13} + 5365250 x^{12} - 14618695 x^{11} + 54147603 x^{10} - 87675235 x^{9} + 296502290 x^{8} - 159474145 x^{7} + 1055587375 x^{6} + 2898777 x^{5} + 3651466705 x^{4} - 1661558095 x^{3} + 7205970510 x^{2} - 5223386330 x + 5105600303 \)

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:  \(77457396436446494932074374685797656495361328125=5^{13}\cdot 6029^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.03$
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:  $5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{15} + \frac{2}{25} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{7}{25}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{11} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{7}{25} a + \frac{2}{5}$, $\frac{1}{325} a^{17} + \frac{6}{325} a^{16} + \frac{4}{325} a^{15} + \frac{2}{65} a^{14} + \frac{1}{65} a^{13} - \frac{8}{325} a^{12} + \frac{2}{325} a^{11} + \frac{8}{325} a^{10} - \frac{3}{65} a^{9} - \frac{8}{65} a^{8} + \frac{6}{13} a^{7} - \frac{12}{65} a^{6} - \frac{21}{65} a^{5} + \frac{4}{13} a^{4} + \frac{14}{65} a^{3} + \frac{147}{325} a^{2} - \frac{73}{325} a + \frac{123}{325}$, $\frac{1}{1625} a^{18} + \frac{2}{1625} a^{17} - \frac{7}{1625} a^{16} - \frac{6}{1625} a^{15} - \frac{4}{65} a^{14} - \frac{28}{1625} a^{13} - \frac{96}{1625} a^{12} + \frac{12}{125} a^{11} + \frac{83}{1625} a^{10} + \frac{6}{65} a^{9} + \frac{49}{325} a^{8} - \frac{54}{325} a^{7} + \frac{1}{325} a^{6} - \frac{11}{25} a^{5} - \frac{92}{325} a^{4} - \frac{3}{1625} a^{3} + \frac{379}{1625} a^{2} - \frac{14}{1625} a + \frac{678}{1625}$, $\frac{1}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{19} + \frac{39309513140986587222727758958141181583709175100800237093735313198747229364239}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{18} - \frac{1340943580287782132544966809434243237298289007825324133484359000009320260934508}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{17} - \frac{60381853390982659293012711604772847135776028074869509895886747988725997965477}{38882019118983420571326238938163576383650204843946673386208073980136082961633915} a^{16} + \frac{201868441109426238039466895561374978327752208724456258712155437277841182991041}{74773113690352731867935074881083800737788855469128218050400142269492467233911375} a^{15} - \frac{1992538143449445812083932476658652903951745507124584127130841962207133629210183}{23708548243282573519101365206197302672957441978016264259882971939107367659532875} a^{14} + \frac{6219951995536500517936106467080955072778909449527341948463074873973906812974336}{74773113690352731867935074881083800737788855469128218050400142269492467233911375} a^{13} + \frac{363664608207731612896623206191695546680816080055322650659216783966356696842234}{7420232656294545910558442545451064195353092527470739195841235492392382244586625} a^{12} + \frac{4952491815680379477320477954434999786543728031548077613129943666695546025764747}{194410095594917102856631194690817881918251024219733366931040369900680414808169575} a^{11} + \frac{2843371958294645601810149160553511068316602283839901731820764552402039526087631}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{10} + \frac{8035365615889732682302135954951761275296718410459114896172737503703124793993199}{194410095594917102856631194690817881918251024219733366931040369900680414808169575} a^{9} + \frac{132368068120548104966261352314112995595481958611885716598644140438278771117178}{364746896050501131063097926249189271891652953507942527075122645217036425531275} a^{8} + \frac{15323127391843274625859496476557925075178424994745320043753250160663767087664193}{194410095594917102856631194690817881918251024219733366931040369900680414808169575} a^{7} - \frac{1872488021592034669564204160154006109392273957750632584347188801194424951778587}{14954622738070546373587014976216760147557771093825643610080028453898493446782275} a^{6} - \frac{50010360791885491965283178506002341469368473607083539588272428297973339549293853}{194410095594917102856631194690817881918251024219733366931040369900680414808169575} a^{5} + \frac{143237760107246758527355159934320058726911019520825067348389857728140552318122227}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{4} + \frac{71815875428251133539387622620783273312896553722174201527553669651102845849530393}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{3} + \frac{71134913045663042697295156656230045934421005623805389178235313161678614317907459}{972050477974585514283155973454089409591255121098666834655201849503402074040847875} a^{2} + \frac{19219419283830567103754767755703948277751248760628983962373940421655558571843393}{194410095594917102856631194690817881918251024219733366931040369900680414808169575} a - \frac{165146156998326590259024916307095717294920554252056759350082494168561339168021304}{972050477974585514283155973454089409591255121098666834655201849503402074040847875}$

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

Class group and class number

$C_{16350}$, which has order $16350$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 285982790609 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{6029}) \), 4.0.181744205.1, 5.5.753625.1, 10.10.124464771269983455453125.1

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 24 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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
6029Data not computed