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Properties

Label 20.0.65823070378...3049.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 401^{15}$
Root discriminant $155.21$
Ramified primes $3, 401$
Class number $52500992$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 82, 5002]$ (GRH)
Galois group $C_5:C_4$ (as 20T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5214568583503, 1482365904611, 3703527579233, 784430792742, 993121367567, 265083906914, 156097776357, 40625770060, 18834291180, 3629444756, 1332452796, 181083914, 59459057, 4226052, 1741913, 31501, 30751, -224, 282, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 282*x^18 - 224*x^17 + 30751*x^16 + 31501*x^15 + 1741913*x^14 + 4226052*x^13 + 59459057*x^12 + 181083914*x^11 + 1332452796*x^10 + 3629444756*x^9 + 18834291180*x^8 + 40625770060*x^7 + 156097776357*x^6 + 265083906914*x^5 + 993121367567*x^4 + 784430792742*x^3 + 3703527579233*x^2 + 1482365904611*x + 5214568583503)
 
gp: K = bnfinit(x^20 - 3*x^19 + 282*x^18 - 224*x^17 + 30751*x^16 + 31501*x^15 + 1741913*x^14 + 4226052*x^13 + 59459057*x^12 + 181083914*x^11 + 1332452796*x^10 + 3629444756*x^9 + 18834291180*x^8 + 40625770060*x^7 + 156097776357*x^6 + 265083906914*x^5 + 993121367567*x^4 + 784430792742*x^3 + 3703527579233*x^2 + 1482365904611*x + 5214568583503, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 282 x^{18} - 224 x^{17} + 30751 x^{16} + 31501 x^{15} + 1741913 x^{14} + 4226052 x^{13} + 59459057 x^{12} + 181083914 x^{11} + 1332452796 x^{10} + 3629444756 x^{9} + 18834291180 x^{8} + 40625770060 x^{7} + 156097776357 x^{6} + 265083906914 x^{5} + 993121367567 x^{4} + 784430792742 x^{3} + 3703527579233 x^{2} + 1482365904611 x + 5214568583503 \)

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:  \(65823070378107273576523817203914035257553049=3^{10}\cdot 401^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.21$
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:  $3, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{2} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} + \frac{1}{6} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{14} - \frac{1}{3} a^{13} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{2274} a^{18} + \frac{30}{379} a^{17} - \frac{11}{379} a^{16} - \frac{4}{379} a^{15} + \frac{51}{758} a^{14} + \frac{245}{758} a^{13} - \frac{277}{758} a^{12} - \frac{101}{379} a^{11} - \frac{1091}{2274} a^{10} + \frac{107}{758} a^{9} - \frac{335}{758} a^{8} + \frac{84}{379} a^{7} - \frac{211}{758} a^{6} - \frac{139}{758} a^{5} + \frac{98}{379} a^{4} + \frac{66}{379} a^{3} + \frac{757}{2274} a^{2} - \frac{75}{379} a - \frac{1}{2}$, $\frac{1}{1225352188151557055633766828228348269407201622804720370352119812237208632664197350793842099099100380894767178728815166} a^{19} + \frac{7742661313331672584865244397214250702266078218045328628919141028748037839948246455028023869005189512626744347709}{1225352188151557055633766828228348269407201622804720370352119812237208632664197350793842099099100380894767178728815166} a^{18} + \frac{33701053687416525376795593504113011885320309302974555317218019336479827615663011240369143442619372780207395165032921}{408450729383852351877922276076116089802400540934906790117373270745736210888065783597947366366366793631589059576271722} a^{17} + \frac{14017440294598402389231704070144671977920606540366124491779519323351529427762003954961821940521662989288312566336679}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{16} + \frac{25074869842506649887271012036544505709344502663730850311644744349072180569292920865519145345815609282645365248384223}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{15} + \frac{68044088194798805664912057561470485317275412843979010999004851175797984552435134329862588951298295824889712374776486}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{14} + \frac{147767657516146547238266600104380993814474841325214354235590067695029581578854261864925854026339210785153650430404323}{408450729383852351877922276076116089802400540934906790117373270745736210888065783597947366366366793631589059576271722} a^{13} - \frac{218427943989215243343536638501355442875700895577232095530829358844785370571544337616176640669606548995110939180660479}{1225352188151557055633766828228348269407201622804720370352119812237208632664197350793842099099100380894767178728815166} a^{12} + \frac{305352073707038806779100847084493456946160194855043369592175664348619363129756039224360128825302134348817910146850651}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{11} - \frac{153460054606828776772235637108624280371471704675310133460584454422313785046078501992399431484620398576221205562846799}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{10} - \frac{156018195158663409867543898613236192939312848034771893765318147161781888770678709781915044133145202452642369620210049}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{9} + \frac{80978832938817341157296027794568595999315849731605706793415787396938507867227949249685870753127496801473850415654719}{204225364691926175938961138038058044901200270467453395058686635372868105444032891798973683183183396815794529788135861} a^{8} - \frac{485583072206382449472490089089173045848367511932004522075507004261164254196780164269142691960990483644064851057111315}{1225352188151557055633766828228348269407201622804720370352119812237208632664197350793842099099100380894767178728815166} a^{7} + \frac{513763981000806543742620031340324075634294127278463154316061677165098975489211861369764455851025114959427737036955}{204225364691926175938961138038058044901200270467453395058686635372868105444032891798973683183183396815794529788135861} a^{6} - \frac{179192305794098023356468566515023628254378110345064404623638195036968322682593264255388435306001277785454448655327341}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{5} + \frac{129657052704673916955604663652353376480611433581876595765613427305584985462265431127350677468521475048887756023736145}{1225352188151557055633766828228348269407201622804720370352119812237208632664197350793842099099100380894767178728815166} a^{4} - \frac{242997061112123985755254972324031271396994487572054264131378427621845295333721348483281131845209642234137228378415033}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{3} - \frac{192479411099345751649429380349898382040254736598700226360818183832132682352972525865957083899608059175011844754963147}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a^{2} - \frac{43255585753096542915340451024248632369241368232708077624706810755210616191014089311213566334491986421341067660401457}{612676094075778527816883414114174134703600811402360185176059906118604316332098675396921049549550190447383589364407583} a - \frac{511688304662088820019240534361420130875053953349192064176000299500232243678898938582043307409546691260848824468654}{1616559614975668938830826950169324893677046995784591517614933789231145953382846109226704616225726096167239022069677}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{82}\times C_{5002}$, which has order $52500992$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:C_4$ (as 20T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $C_5:C_4$
Character table for $C_5:C_4$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.0.580330809.1, 5.5.160801.1 x5, 10.10.10368641602001.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed