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Properties

Label 20.0.29832890652...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $265.30$
Ramified primes $2, 3, 5, 11$
Class number $562707872$ (GRH)
Class group $[2, 4, 70338484]$ (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![656014524319351, -83694270864252, 203877383312922, -23239353719744, 29048118383191, -2934443802712, 2493011581014, -220520041752, 142386950259, -10839365132, 5641661940, -360471336, 156710384, -8096720, 3009194, -118404, 38215, -1024, 290, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 290*x^18 - 1024*x^17 + 38215*x^16 - 118404*x^15 + 3009194*x^14 - 8096720*x^13 + 156710384*x^12 - 360471336*x^11 + 5641661940*x^10 - 10839365132*x^9 + 142386950259*x^8 - 220520041752*x^7 + 2493011581014*x^6 - 2934443802712*x^5 + 29048118383191*x^4 - 23239353719744*x^3 + 203877383312922*x^2 - 83694270864252*x + 656014524319351)
 
gp: K = bnfinit(x^20 - 4*x^19 + 290*x^18 - 1024*x^17 + 38215*x^16 - 118404*x^15 + 3009194*x^14 - 8096720*x^13 + 156710384*x^12 - 360471336*x^11 + 5641661940*x^10 - 10839365132*x^9 + 142386950259*x^8 - 220520041752*x^7 + 2493011581014*x^6 - 2934443802712*x^5 + 29048118383191*x^4 - 23239353719744*x^3 + 203877383312922*x^2 - 83694270864252*x + 656014524319351, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 290 x^{18} - 1024 x^{17} + 38215 x^{16} - 118404 x^{15} + 3009194 x^{14} - 8096720 x^{13} + 156710384 x^{12} - 360471336 x^{11} + 5641661940 x^{10} - 10839365132 x^{9} + 142386950259 x^{8} - 220520041752 x^{7} + 2493011581014 x^{6} - 2934443802712 x^{5} + 29048118383191 x^{4} - 23239353719744 x^{3} + 203877383312922 x^{2} - 83694270864252 x + 656014524319351 \)

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:  \(2983289065263288625233938941476864000000000000000=2^{55}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $265.30$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2640=2^{4}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2640}(1,·)$, $\chi_{2640}(2627,·)$, $\chi_{2640}(961,·)$, $\chi_{2640}(1609,·)$, $\chi_{2640}(203,·)$, $\chi_{2640}(1681,·)$, $\chi_{2640}(467,·)$, $\chi_{2640}(1643,·)$, $\chi_{2640}(1849,·)$, $\chi_{2640}(1369,·)$, $\chi_{2640}(2401,·)$, $\chi_{2640}(2363,·)$, $\chi_{2640}(169,·)$, $\chi_{2640}(683,·)$, $\chi_{2640}(2161,·)$, $\chi_{2640}(947,·)$, $\chi_{2640}(707,·)$, $\chi_{2640}(889,·)$, $\chi_{2640}(1907,·)$, $\chi_{2640}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{2}{9} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{14} - \frac{1}{9} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{911629726107703092577066842729} a^{18} + \frac{12174865354524928291620592534}{911629726107703092577066842729} a^{17} - \frac{15300731719173862225192127332}{911629726107703092577066842729} a^{16} + \frac{12680389100514002625755354696}{303876575369234364192355614243} a^{15} - \frac{141472593619456960387277147773}{911629726107703092577066842729} a^{14} + \frac{131300471906280338326073269675}{911629726107703092577066842729} a^{13} + \frac{12095242995742626338338302874}{911629726107703092577066842729} a^{12} + \frac{135140937007994623109351759942}{911629726107703092577066842729} a^{11} - \frac{121170216831039280496752722932}{911629726107703092577066842729} a^{10} - \frac{424342951412091965367441094504}{911629726107703092577066842729} a^{9} + \frac{127729085974590892521053690260}{303876575369234364192355614243} a^{8} + \frac{31050968505518320228771728228}{101292191789744788064118538081} a^{7} - \frac{49657000149383561606700320077}{303876575369234364192355614243} a^{6} + \frac{97663859908032608921770432769}{303876575369234364192355614243} a^{5} + \frac{150456847897354943000450425247}{303876575369234364192355614243} a^{4} + \frac{380238021289904869949657501447}{911629726107703092577066842729} a^{3} - \frac{382383062667520529612312633977}{911629726107703092577066842729} a^{2} - \frac{203212586403106625502502684562}{911629726107703092577066842729} a - \frac{10581332432512457111586151693}{21200691304830304478536438203}$, $\frac{1}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{19} - \frac{64000861034180157326720923857270644478125}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{18} + \frac{3469571027923080500488080644016698341494022466219413065349273198703711}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{17} - \frac{2349302970029251230785151568310529556827534173221251692611316098892708}{88952600873009131727347975881602515802160277027641592063434751484518803} a^{16} + \frac{33733983800946837257883563597293154395209313984431260513825223202782}{689555045527202571529829270400019502342327728896446450104145360345107} a^{15} + \frac{12005384727387578951778337900531431455043273279526720268905969022183539}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{14} + \frac{310106837690305968057412752791364155234644270246424979139746150743122}{6205995409744823143768463433600175521080949560068018050937308243105963} a^{13} - \frac{26172925115040256236504538665110273465466443185306879460030311990461647}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{12} - \frac{47233200374178224383932337781542336348355591060856608759056706124692}{3982952277895931271373789964847873841887773598252608599855287379903827} a^{11} - \frac{41125123078668779245636393610494332058610370564494914709333798190719422}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{10} - \frac{125638001900814396380429245465549023642069228125302255669687828806347362}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{9} - \frac{38955427182964178958098790677140289330362769521174214741872120095314326}{88952600873009131727347975881602515802160277027641592063434751484518803} a^{8} - \frac{91637703397595597742325593222128123873954336757874276979988951154069}{442550253099547919041532218316430426876419288694734288872809708878203} a^{7} + \frac{18964834799452461679377108196066284940909370207340812288978341507376644}{88952600873009131727347975881602515802160277027641592063434751484518803} a^{6} - \frac{33608003714157158182624027640196668056944323850470564129438597333504884}{88952600873009131727347975881602515802160277027641592063434751484518803} a^{5} + \frac{10923895158569938544243671514523428595718396328585184997484662041567662}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{4} + \frac{128602049066184820542746434196025266059023464920018356328356220703895302}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{3} + \frac{45358552049808839653343352949414123465290408468298042582826310917571169}{266857802619027395182043927644807547406480831082924776190304254453556409} a^{2} + \frac{24307853418160963290884832351566356540642821899354655628132590854907776}{266857802619027395182043927644807547406480831082924776190304254453556409} a - \frac{1660778447676298618045026015966432783900847667757005650107069183358471}{6205995409744823143768463433600175521080949560068018050937308243105963}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{70338484}$, which has order $562707872$ (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:  \( 14452469.589232503 \) (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

\(\Q(\sqrt{10}) \), 4.0.2304000.2, \(\Q(\zeta_{11})^+\), 10.10.21950349414400000.1

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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$3.10.5.1$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}$
5Data not computed
11Data not computed