↑

Properties

Label 20.0.10703759836...6889.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 241^{19}$
Root discriminant $317.30$
Ramified primes $3, 241$
Class number $4197873844$ (GRH)
Class group $[2, 2098936922]$ (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3630610259968, -1703567371264, 1273319969280, -468112596032, 235967317824, -83554994512, 39306798768, -12300643612, 4604716892, -1060770227, 298062451, -46507613, 10909277, -1081006, 283662, -18838, 6998, -255, 127, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 127*x^18 - 255*x^17 + 6998*x^16 - 18838*x^15 + 283662*x^14 - 1081006*x^13 + 10909277*x^12 - 46507613*x^11 + 298062451*x^10 - 1060770227*x^9 + 4604716892*x^8 - 12300643612*x^7 + 39306798768*x^6 - 83554994512*x^5 + 235967317824*x^4 - 468112596032*x^3 + 1273319969280*x^2 - 1703567371264*x + 3630610259968)
 
gp: K = bnfinit(x^20 - x^19 + 127*x^18 - 255*x^17 + 6998*x^16 - 18838*x^15 + 283662*x^14 - 1081006*x^13 + 10909277*x^12 - 46507613*x^11 + 298062451*x^10 - 1060770227*x^9 + 4604716892*x^8 - 12300643612*x^7 + 39306798768*x^6 - 83554994512*x^5 + 235967317824*x^4 - 468112596032*x^3 + 1273319969280*x^2 - 1703567371264*x + 3630610259968, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 127 x^{18} - 255 x^{17} + 6998 x^{16} - 18838 x^{15} + 283662 x^{14} - 1081006 x^{13} + 10909277 x^{12} - 46507613 x^{11} + 298062451 x^{10} - 1060770227 x^{9} + 4604716892 x^{8} - 12300643612 x^{7} + 39306798768 x^{6} - 83554994512 x^{5} + 235967317824 x^{4} - 468112596032 x^{3} + 1273319969280 x^{2} - 1703567371264 x + 3630610259968 \)

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:  \(107037598360768458712497472964188759976745670736889=3^{10}\cdot 241^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $317.30$
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:  $3, 241$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(723=3\cdot 241\)
Dirichlet character group:    $\lbrace$$\chi_{723}(1,·)$, $\chi_{723}(580,·)$, $\chi_{723}(391,·)$, $\chi_{723}(328,·)$, $\chi_{723}(266,·)$, $\chi_{723}(205,·)$, $\chi_{723}(659,·)$, $\chi_{723}(277,·)$, $\chi_{723}(281,·)$, $\chi_{723}(154,·)$, $\chi_{723}(91,·)$, $\chi_{723}(476,·)$, $\chi_{723}(481,·)$, $\chi_{723}(347,·)$, $\chi_{723}(625,·)$, $\chi_{723}(488,·)$, $\chi_{723}(617,·)$, $\chi_{723}(683,·)$, $\chi_{723}(305,·)$, $\chi_{723}(698,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} + \frac{3}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} - \frac{3}{128} a^{4} + \frac{3}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{11} - \frac{1}{256} a^{9} - \frac{5}{256} a^{7} - \frac{7}{256} a^{5} + \frac{3}{64} a^{3}$, $\frac{1}{1024} a^{12} - \frac{1}{512} a^{11} + \frac{1}{1024} a^{10} - \frac{1}{512} a^{9} - \frac{1}{1024} a^{8} - \frac{15}{512} a^{7} - \frac{5}{1024} a^{6} - \frac{27}{512} a^{5} + \frac{1}{256} a^{4} - \frac{5}{128} a^{3} + \frac{1}{8} a$, $\frac{1}{4096} a^{13} + \frac{1}{4096} a^{12} - \frac{5}{4096} a^{11} - \frac{7}{4096} a^{10} + \frac{1}{4096} a^{9} + \frac{15}{4096} a^{8} + \frac{49}{4096} a^{7} + \frac{51}{4096} a^{6} + \frac{53}{2048} a^{5} + \frac{17}{1024} a^{4} - \frac{67}{512} a^{3} + \frac{7}{32} a^{2} + \frac{3}{32} a - \frac{1}{4}$, $\frac{1}{8192} a^{14} - \frac{1}{8192} a^{13} - \frac{3}{8192} a^{12} - \frac{5}{8192} a^{11} - \frac{13}{8192} a^{10} - \frac{27}{8192} a^{9} + \frac{15}{8192} a^{8} - \frac{231}{8192} a^{7} - \frac{11}{512} a^{6} + \frac{47}{1024} a^{5} - \frac{13}{512} a^{4} + \frac{9}{512} a^{3} + \frac{3}{64} a^{2} + \frac{7}{32} a - \frac{1}{4}$, $\frac{1}{16384} a^{15} - \frac{1}{4096} a^{12} - \frac{3}{8192} a^{11} - \frac{1}{4096} a^{10} - \frac{5}{2048} a^{9} - \frac{7}{4096} a^{8} + \frac{141}{16384} a^{7} + \frac{117}{4096} a^{6} + \frac{23}{1024} a^{5} + \frac{61}{1024} a^{4} + \frac{147}{1024} a^{3} - \frac{27}{128} a^{2} - \frac{11}{64} a + \frac{1}{8}$, $\frac{1}{32768} a^{16} - \frac{1}{8192} a^{13} + \frac{5}{16384} a^{12} + \frac{7}{8192} a^{11} + \frac{5}{4096} a^{10} - \frac{15}{8192} a^{9} - \frac{3}{32768} a^{8} - \frac{51}{8192} a^{7} + \frac{27}{1024} a^{6} + \frac{47}{2048} a^{5} + \frac{55}{2048} a^{4} - \frac{1}{64} a^{3} + \frac{25}{128} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1048576} a^{17} - \frac{7}{1048576} a^{16} - \frac{5}{524288} a^{15} + \frac{5}{262144} a^{14} + \frac{31}{524288} a^{13} + \frac{195}{524288} a^{12} + \frac{207}{131072} a^{11} + \frac{411}{262144} a^{10} - \frac{1895}{1048576} a^{9} + \frac{2553}{1048576} a^{8} + \frac{12509}{524288} a^{7} + \frac{719}{131072} a^{6} - \frac{657}{32768} a^{5} + \frac{3257}{65536} a^{4} + \frac{6681}{32768} a^{3} - \frac{153}{1024} a^{2} + \frac{343}{2048} a + \frac{87}{256}$, $\frac{1}{2097152} a^{18} + \frac{5}{2097152} a^{16} - \frac{25}{1048576} a^{15} - \frac{27}{1048576} a^{14} - \frac{25}{262144} a^{13} + \frac{337}{1048576} a^{12} - \frac{787}{524288} a^{11} + \frac{2189}{2097152} a^{10} + \frac{965}{262144} a^{9} + \frac{3017}{2097152} a^{8} - \frac{5817}{1048576} a^{7} - \frac{5403}{262144} a^{6} - \frac{1397}{131072} a^{5} + \frac{1281}{131072} a^{4} - \frac{13809}{65536} a^{3} + \frac{537}{4096} a^{2} - \frac{1383}{4096} a - \frac{31}{512}$, $\frac{1}{255807369130130041412154781350209826498582127104226495698660340269056} a^{19} + \frac{1540204541979208131545891831434822666311587041861052096185373}{7993980285316563794129836917194057078080691472007077990583135633408} a^{18} - \frac{66142912283065183740913001547133301593824076427811896068194837}{255807369130130041412154781350209826498582127104226495698660340269056} a^{17} + \frac{703878205784115643494655574383223077510129841704496827156503033}{63951842282532510353038695337552456624645531776056623924665085067264} a^{16} - \frac{3453988026816963808037125305773242461029262387974540003064340473}{127903684565065020706077390675104913249291063552113247849330170134528} a^{15} + \frac{786144297911273717733070952330915061693455786985220681198209959}{15987960570633127588259673834388114156161382944014155981166271266816} a^{14} + \frac{6036732817540151381187724078098991153328526955436502114921561771}{127903684565065020706077390675104913249291063552113247849330170134528} a^{13} + \frac{6049850772552377724349439926834649305834853577067427415325199835}{31975921141266255176519347668776228312322765888028311962332542533632} a^{12} + \frac{341012263798313570507716658603938741128191280911615123811284189149}{255807369130130041412154781350209826498582127104226495698660340269056} a^{11} - \frac{2668870321205425804716097921780385382461104443458762309994930155}{15987960570633127588259673834388114156161382944014155981166271266816} a^{10} - \frac{964344245916180695663422468332453417462009344268527702748334307009}{255807369130130041412154781350209826498582127104226495698660340269056} a^{9} - \frac{87905141206596398312002376907045729145090097713007715688921923511}{63951842282532510353038695337552456624645531776056623924665085067264} a^{8} + \frac{1007385021063645651445114872315650849828272485299683398683146564705}{63951842282532510353038695337552456624645531776056623924665085067264} a^{7} - \frac{37585552635487048399363644659353462370004832356202567124995597129}{1998495071329140948532459229298514269520172868001769497645783908352} a^{6} + \frac{472414463260585654480528476433879102187138487031636661367739080469}{15987960570633127588259673834388114156161382944014155981166271266816} a^{5} + \frac{5816635891197667080656376795709127387899339994133486390098038093}{3996990142658281897064918458597028539040345736003538995291567816704} a^{4} + \frac{286202746399271561667350750058324979380840795929190026864980617555}{3996990142658281897064918458597028539040345736003538995291567816704} a^{3} + \frac{115823718192556811591359165522604764894914604597279264977525683405}{499623767832285237133114807324628567380043217000442374411445977088} a^{2} - \frac{11989431466287997367025709020396953950403614969274718776848925495}{249811883916142618566557403662314283690021608500221187205722988544} a + \frac{13386605376100740798529042978578861154614050231730007548675075781}{31226485489517827320819675457789285461252701062527648400715373568}$

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

Class group and class number

$C_{2}\times C_{2098936922}$, which has order $4197873844$ (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:  \( 7201319699389.811 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{241}) \), 4.0.125977689.1, 5.5.3373402561.1, 10.10.2742542606093287451761.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
241Data not computed