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Properties

Label 20.0.38319164111...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 19^{10}$
Root discriminant $190.18$
Ramified primes $2, 5, 19$
Class number $11652528$ (GRH)
Class group $[11652528]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![448168676849, 37137063760, 182832246360, 10079684960, 36060966430, 1783921248, 4538700640, 224982100, 396512975, 18779080, 25483210, 1026700, 1241510, 25260, 45910, 36, 1360, -20, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 30*x^18 - 20*x^17 + 1360*x^16 + 36*x^15 + 45910*x^14 + 25260*x^13 + 1241510*x^12 + 1026700*x^11 + 25483210*x^10 + 18779080*x^9 + 396512975*x^8 + 224982100*x^7 + 4538700640*x^6 + 1783921248*x^5 + 36060966430*x^4 + 10079684960*x^3 + 182832246360*x^2 + 37137063760*x + 448168676849)
 
gp: K = bnfinit(x^20 + 30*x^18 - 20*x^17 + 1360*x^16 + 36*x^15 + 45910*x^14 + 25260*x^13 + 1241510*x^12 + 1026700*x^11 + 25483210*x^10 + 18779080*x^9 + 396512975*x^8 + 224982100*x^7 + 4538700640*x^6 + 1783921248*x^5 + 36060966430*x^4 + 10079684960*x^3 + 182832246360*x^2 + 37137063760*x + 448168676849, 1)
 

Normalized defining polynomial

\( x^{20} + 30 x^{18} - 20 x^{17} + 1360 x^{16} + 36 x^{15} + 45910 x^{14} + 25260 x^{13} + 1241510 x^{12} + 1026700 x^{11} + 25483210 x^{10} + 18779080 x^{9} + 396512975 x^{8} + 224982100 x^{7} + 4538700640 x^{6} + 1783921248 x^{5} + 36060966430 x^{4} + 10079684960 x^{3} + 182832246360 x^{2} + 37137063760 x + 448168676849 \)

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:  \(3831916411125625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $190.18$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3800=2^{3}\cdot 5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3800}(1,·)$, $\chi_{3800}(3269,·)$, $\chi_{3800}(2849,·)$, $\chi_{3800}(1101,·)$, $\chi_{3800}(2509,·)$, $\chi_{3800}(341,·)$, $\chi_{3800}(569,·)$, $\chi_{3800}(3609,·)$, $\chi_{3800}(989,·)$, $\chi_{3800}(1861,·)$, $\chi_{3800}(3041,·)$, $\chi_{3800}(229,·)$, $\chi_{3800}(1521,·)$, $\chi_{3800}(2089,·)$, $\chi_{3800}(1329,·)$, $\chi_{3800}(3381,·)$, $\chi_{3800}(2281,·)$, $\chi_{3800}(761,·)$, $\chi_{3800}(2621,·)$, $\chi_{3800}(1749,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{50} a^{10} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{9}{25} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{7}{50}$, $\frac{1}{50} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{9}{25} a^{6} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{7}{50} a$, $\frac{1}{350} a^{12} - \frac{3}{350} a^{11} - \frac{3}{350} a^{10} - \frac{13}{35} a^{9} + \frac{9}{35} a^{8} + \frac{29}{175} a^{7} + \frac{11}{350} a^{6} - \frac{29}{350} a^{5} + \frac{33}{70} a^{4} - \frac{167}{350} a^{2} - \frac{59}{350} a - \frac{149}{350}$, $\frac{1}{350} a^{13} + \frac{1}{175} a^{11} + \frac{1}{350} a^{10} - \frac{9}{35} a^{9} - \frac{81}{175} a^{8} - \frac{19}{70} a^{7} - \frac{47}{175} a^{6} - \frac{66}{175} a^{5} - \frac{13}{70} a^{4} - \frac{27}{350} a^{3} - \frac{1}{5} a^{2} - \frac{37}{175} a - \frac{27}{350}$, $\frac{1}{350} a^{14} + \frac{12}{25} a^{9} + \frac{1}{70} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{50} a^{4} - \frac{2}{35} a^{2} - \frac{1}{5} a + \frac{6}{35}$, $\frac{1}{2100} a^{15} - \frac{1}{1050} a^{13} - \frac{1}{700} a^{12} - \frac{1}{1050} a^{11} - \frac{1}{100} a^{10} + \frac{59}{420} a^{9} + \frac{79}{350} a^{8} + \frac{253}{1050} a^{7} - \frac{881}{2100} a^{6} + \frac{281}{2100} a^{5} + \frac{137}{420} a^{4} - \frac{149}{525} a^{3} - \frac{59}{2100} a^{2} - \frac{3}{50} a + \frac{823}{2100}$, $\frac{1}{14700} a^{16} + \frac{1}{7350} a^{15} - \frac{1}{735} a^{14} + \frac{11}{14700} a^{13} - \frac{1}{7350} a^{12} - \frac{19}{2100} a^{11} + \frac{17}{2940} a^{10} - \frac{31}{147} a^{9} + \frac{1532}{3675} a^{8} + \frac{6749}{14700} a^{7} + \frac{773}{2940} a^{6} + \frac{6383}{14700} a^{5} + \frac{82}{245} a^{4} - \frac{2259}{4900} a^{3} - \frac{1493}{7350} a^{2} - \frac{6953}{14700} a - \frac{1919}{7350}$, $\frac{1}{14700} a^{17} - \frac{1}{4900} a^{15} + \frac{3}{4900} a^{14} + \frac{3}{2450} a^{13} + \frac{3}{2450} a^{12} + \frac{47}{4900} a^{11} - \frac{47}{4900} a^{10} + \frac{3067}{14700} a^{9} - \frac{5171}{14700} a^{8} - \frac{1209}{4900} a^{7} - \frac{907}{2450} a^{6} - \frac{1061}{2940} a^{5} + \frac{556}{1225} a^{4} - \frac{1054}{3675} a^{3} + \frac{59}{490} a^{2} + \frac{243}{2450} a + \frac{4169}{14700}$, $\frac{1}{328698637326345300} a^{18} - \frac{3925689864589}{164349318663172650} a^{17} + \frac{357700644121}{54783106221057550} a^{16} - \frac{21155447299549}{164349318663172650} a^{15} - \frac{116156835351}{559011287969975} a^{14} - \frac{250599348971309}{328698637326345300} a^{13} - \frac{13165549252369}{54783106221057550} a^{12} + \frac{102778190592142}{11739237047369475} a^{11} - \frac{2397415427009567}{328698637326345300} a^{10} - \frac{16498349960580779}{54783106221057550} a^{9} - \frac{33942869550836123}{328698637326345300} a^{8} + \frac{145700450240992091}{328698637326345300} a^{7} + \frac{50381306781952239}{109566212442115100} a^{6} + \frac{12679675165865668}{82174659331586325} a^{5} + \frac{23193548423839987}{109566212442115100} a^{4} + \frac{41005317947859483}{109566212442115100} a^{3} + \frac{13711540747937807}{65739727465269060} a^{2} + \frac{12636221926050521}{82174659331586325} a - \frac{52300006297717623}{109566212442115100}$, $\frac{1}{16948589957275632100711567362963516864483706603725300} a^{19} - \frac{1512970968308630464399813772698676}{1412382496439636008392630613580293072040308883643775} a^{18} - \frac{30179885125347036325442835142873817351793101004}{1412382496439636008392630613580293072040308883643775} a^{17} - \frac{58865054894465324902555107728335809664176859287}{2824764992879272016785261227160586144080617767287550} a^{16} - \frac{491138327304914113705547021771231130051360853931}{2421227136753661728673081051851930980640529514817900} a^{15} - \frac{1413402859332190474947163944288370570100998392173}{5649529985758544033570522454321172288161235534575100} a^{14} - \frac{229836397379770088271540802338923304446853958964}{605306784188415432168270262962982745160132378704475} a^{13} + \frac{89950964116787447766642769185974777008257490951}{807075712251220576224360350617310326880176504939300} a^{12} - \frac{62902401258174010253095206153020481864115979062291}{16948589957275632100711567362963516864483706603725300} a^{11} + \frac{17126559839573187851354727906237279259665406948753}{2421227136753661728673081051851930980640529514817900} a^{10} - \frac{948790282957774217776895278654108225832764274657539}{8474294978637816050355783681481758432241853301862650} a^{9} + \frac{1866047795784529236276672968688843659152754467059967}{5649529985758544033570522454321172288161235534575100} a^{8} + \frac{4591054785298008565619878877107671870790939690277581}{16948589957275632100711567362963516864483706603725300} a^{7} - \frac{5741624959018841549660193969621674241544110597557021}{16948589957275632100711567362963516864483706603725300} a^{6} - \frac{290463341275384635174017071859252634535832767063906}{847429497863781605035578368148175843224185330186265} a^{5} - \frac{397010018374623225310117214174947628499187991069143}{8474294978637816050355783681481758432241853301862650} a^{4} - \frac{746225723425769531290410278085917928710767912498613}{16948589957275632100711567362963516864483706603725300} a^{3} + \frac{381084135078238979705109731216973199521154236560243}{1129905997151708806714104490864234457632247106915020} a^{2} + \frac{162308623967163081483625994324622721857527734559851}{1129905997151708806714104490864234457632247106915020} a - \frac{46136664100143000091893079284712208559977657999617}{158398036983884412156182872551060905275548659847900}$

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

Class group and class number

$C_{11652528}$, which has order $11652528$ (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:  \( 19344397.966990974 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{10}, \sqrt{-38})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.1889113616943359375.3, 10.0.12380495000000000000000.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
5Data not computed
19Data not computed