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Properties

Label 20.0.82802905234...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{35}\cdot 11^{16}$
Root discriminant $394.37$
Ramified primes $2, 3, 5, 11$
Class number $1043368000$ (GRH)
Class group $[2, 2, 2, 10, 10, 1304210]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![70536147920701, -27446286638760, 3143847094100, 1619481226110, 506495275550, -137024708502, -32642132485, 15845595150, 4830793285, -626326800, -333632526, 13904880, 16919275, -110880, -539050, -198, 11495, 0, -145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 145*x^18 + 11495*x^16 - 198*x^15 - 539050*x^14 - 110880*x^13 + 16919275*x^12 + 13904880*x^11 - 333632526*x^10 - 626326800*x^9 + 4830793285*x^8 + 15845595150*x^7 - 32642132485*x^6 - 137024708502*x^5 + 506495275550*x^4 + 1619481226110*x^3 + 3143847094100*x^2 - 27446286638760*x + 70536147920701)
 
gp: K = bnfinit(x^20 - 145*x^18 + 11495*x^16 - 198*x^15 - 539050*x^14 - 110880*x^13 + 16919275*x^12 + 13904880*x^11 - 333632526*x^10 - 626326800*x^9 + 4830793285*x^8 + 15845595150*x^7 - 32642132485*x^6 - 137024708502*x^5 + 506495275550*x^4 + 1619481226110*x^3 + 3143847094100*x^2 - 27446286638760*x + 70536147920701, 1)
 

Normalized defining polynomial

\( x^{20} - 145 x^{18} + 11495 x^{16} - 198 x^{15} - 539050 x^{14} - 110880 x^{13} + 16919275 x^{12} + 13904880 x^{11} - 333632526 x^{10} - 626326800 x^{9} + 4830793285 x^{8} + 15845595150 x^{7} - 32642132485 x^{6} - 137024708502 x^{5} + 506495275550 x^{4} + 1619481226110 x^{3} + 3143847094100 x^{2} - 27446286638760 x + 70536147920701 \)

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:  \(8280290523419410812039184570312500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{35}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $394.37$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(2027,·)$, $\chi_{3300}(1607,·)$, $\chi_{3300}(587,·)$, $\chi_{3300}(467,·)$, $\chi_{3300}(2809,·)$, $\chi_{3300}(1369,·)$, $\chi_{3300}(2183,·)$, $\chi_{3300}(289,·)$, $\chi_{3300}(229,·)$, $\chi_{3300}(1703,·)$, $\chi_{3300}(647,·)$, $\chi_{3300}(2941,·)$, $\chi_{3300}(2963,·)$, $\chi_{3300}(181,·)$, $\chi_{3300}(2423,·)$, $\chi_{3300}(1849,·)$, $\chi_{3300}(1343,·)$, $\chi_{3300}(1021,·)$, $\chi_{3300}(3061,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{10} + \frac{2}{5} a^{7} - \frac{1}{3} a^{6} - \frac{2}{5} a^{5} - \frac{1}{15} a^{2} + \frac{1}{15}$, $\frac{1}{75} a^{13} - \frac{1}{75} a^{12} + \frac{2}{75} a^{11} + \frac{7}{75} a^{10} + \frac{1}{5} a^{9} + \frac{7}{25} a^{8} + \frac{19}{75} a^{7} + \frac{2}{75} a^{6} + \frac{9}{25} a^{5} - \frac{2}{5} a^{4} - \frac{16}{75} a^{3} - \frac{14}{75} a^{2} - \frac{2}{75} a - \frac{22}{75}$, $\frac{1}{75} a^{14} + \frac{1}{75} a^{12} - \frac{2}{25} a^{11} + \frac{7}{75} a^{10} + \frac{12}{25} a^{9} - \frac{7}{15} a^{8} + \frac{7}{25} a^{7} + \frac{14}{75} a^{6} - \frac{6}{25} a^{5} + \frac{29}{75} a^{4} - \frac{2}{5} a^{3} - \frac{16}{75} a^{2} - \frac{3}{25} a - \frac{7}{75}$, $\frac{1}{75} a^{15} + \frac{1}{15} a^{11} - \frac{2}{25} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{2}{5} a^{6} + \frac{17}{75} a^{5} - \frac{1}{15} a - \frac{6}{25}$, $\frac{1}{150975} a^{16} + \frac{4}{1525} a^{15} + \frac{677}{150975} a^{14} + \frac{23}{4575} a^{13} - \frac{1937}{150975} a^{12} + \frac{21}{305} a^{11} + \frac{69}{1525} a^{10} - \frac{582}{1525} a^{9} - \frac{15527}{50325} a^{8} + \frac{326}{4575} a^{7} + \frac{18811}{50325} a^{6} - \frac{1982}{4575} a^{5} - \frac{44987}{150975} a^{4} + \frac{1637}{4575} a^{3} + \frac{28922}{150975} a^{2} + \frac{499}{1525} a - \frac{5089}{30195}$, $\frac{1}{150975} a^{17} + \frac{35}{6039} a^{15} + \frac{4}{1525} a^{14} - \frac{551}{150975} a^{13} - \frac{16}{1525} a^{12} - \frac{92}{1525} a^{11} + \frac{7}{915} a^{10} - \frac{17078}{50325} a^{9} + \frac{281}{1525} a^{8} - \frac{10262}{50325} a^{7} - \frac{736}{4575} a^{6} - \frac{5473}{30195} a^{5} - \frac{992}{4575} a^{4} + \frac{44861}{150975} a^{3} + \frac{8}{305} a^{2} + \frac{74743}{150975} a + \frac{434}{915}$, $\frac{1}{150975} a^{18} + \frac{4}{4575} a^{15} + \frac{101}{16775} a^{14} + \frac{6}{1525} a^{13} + \frac{2899}{150975} a^{12} + \frac{84}{1525} a^{11} + \frac{64}{16775} a^{10} + \frac{1153}{4575} a^{9} - \frac{11216}{50325} a^{8} + \frac{421}{1525} a^{7} + \frac{57196}{150975} a^{6} - \frac{911}{4575} a^{5} - \frac{5146}{16775} a^{4} + \frac{1489}{4575} a^{3} - \frac{3026}{16775} a^{2} - \frac{199}{1525} a + \frac{44881}{150975}$, $\frac{1}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{19} - \frac{5334083995888968460702980180060826731808944246523840187332930326071535919246538092740876757}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{18} + \frac{35068750404687300793498734762372125324545323150187530244331830917206124557486165226050636411}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{17} + \frac{36820353386262058871317120917426140337099776561408111102926437624243466358093596519078673803}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{16} + \frac{45102004941194690060601443074444042728812724835481943018588953639214053721686529313639681731571}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{15} - \frac{61434660952402226702759188709237559173644117725258141820921536982616202765048641380053265944199}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{14} - \frac{1046357030655117272278937508678793045828650717148398258021051884072995387154516481089670903902}{256241632650078848805755793583849904020419442225238285982084718358110335670932931391560220957755} a^{13} + \frac{101053475825442879043828113079955629734671632786940107359709157069988455758169720449473926632531}{3843624489751182732086336903757748560306291633378574289731270775371655035063993970873403314366325} a^{12} + \frac{223409107306852745898758398327599836771735505218711243862190153631082032186918054000549181705317}{3843624489751182732086336903757748560306291633378574289731270775371655035063993970873403314366325} a^{11} + \frac{225470164922319069477043333323351313510350516354485757332498483026042901532962908226221859558408}{3843624489751182732086336903757748560306291633378574289731270775371655035063993970873403314366325} a^{10} + \frac{1212449501943050722842265810983169990533729548210802699123332903308948009219699506073054553172688}{3843624489751182732086336903757748560306291633378574289731270775371655035063993970873403314366325} a^{9} - \frac{244984306669338637664092992968960666304261446454675302380471796772339966001701769205780144532104}{1281208163250394244028778967919249520102097211126191429910423591790551678354664656957801104788775} a^{8} - \frac{50547011404918087584598862620619228927031985826809340002619581017310924095307427878717240700089}{1048261224477595290569000973752113243719897718194156624472164756919542282290180173874564540281725} a^{7} + \frac{85220345432833567484287573584301059787106336199355837983203444576740687349045379687846891393272}{209652244895519058113800194750422648743979543638831324894432951383908456458036034774912908056345} a^{6} - \frac{500440284775765663840706642943610580694354400652351562336008601174691778345730788713068592372928}{1048261224477595290569000973752113243719897718194156624472164756919542282290180173874564540281725} a^{5} - \frac{34424415957960639255012148496460323880872799598479999025480973294459863648858442753101188991106}{209652244895519058113800194750422648743979543638831324894432951383908456458036034774912908056345} a^{4} - \frac{4411312034260713595790150330121164369425972347612529891456582457602177552051916704800916079284472}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{3} + \frac{5446642864775913130769766549364102642523621694690886242462992107444729266428001420256164356201018}{11530873469253548196259010711273245680918874900135722869193812326114965105191981912620209943098975} a^{2} - \frac{752156749592157268185936997119048707467417665530106047579880652863412674906374784845888901346376}{3843624489751182732086336903757748560306291633378574289731270775371655035063993970873403314366325} a + \frac{245122767869658179996621921685343987574099658052796273868767024867742474791711386634639595333}{2368222113216994905783325264176061959523285048292405600573795918281980921173132452787063040275}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{10}\times C_{1304210}$, which has order $1043368000$ (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:  \( 864758391.9354854 \) (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

\(\Q(\sqrt{5}) \), 4.0.18000.1, 5.5.5719140625.2, 10.10.163542847442626953125.1

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 R R $20$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$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$$[\ ]_{2}^{2}$
5Data not computed
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$