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Properties

Label 20.0.25339956687...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 41^{19}$
Root discriminant $131.88$
Ramified primes $3, 5, 41$
Class number $2089216$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 32644]$ (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![3942396001, 1091380322, 2035349497, 248531503, 104284815, 527395272, -87440612, 20117476, 59475307, -17477774, 21468861, -3113921, 3027292, -225104, 216784, -8172, 8025, -146, 145, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 145*x^18 - 146*x^17 + 8025*x^16 - 8172*x^15 + 216784*x^14 - 225104*x^13 + 3027292*x^12 - 3113921*x^11 + 21468861*x^10 - 17477774*x^9 + 59475307*x^8 + 20117476*x^7 - 87440612*x^6 + 527395272*x^5 + 104284815*x^4 + 248531503*x^3 + 2035349497*x^2 + 1091380322*x + 3942396001)
 
gp: K = bnfinit(x^20 - x^19 + 145*x^18 - 146*x^17 + 8025*x^16 - 8172*x^15 + 216784*x^14 - 225104*x^13 + 3027292*x^12 - 3113921*x^11 + 21468861*x^10 - 17477774*x^9 + 59475307*x^8 + 20117476*x^7 - 87440612*x^6 + 527395272*x^5 + 104284815*x^4 + 248531503*x^3 + 2035349497*x^2 + 1091380322*x + 3942396001, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 145 x^{18} - 146 x^{17} + 8025 x^{16} - 8172 x^{15} + 216784 x^{14} - 225104 x^{13} + 3027292 x^{12} - 3113921 x^{11} + 21468861 x^{10} - 17477774 x^{9} + 59475307 x^{8} + 20117476 x^{7} - 87440612 x^{6} + 527395272 x^{5} + 104284815 x^{4} + 248531503 x^{3} + 2035349497 x^{2} + 1091380322 x + 3942396001 \)

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:  \(2533995668777081618288969459861002822265625=3^{10}\cdot 5^{10}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.88$
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:  $3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(615=3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{615}(256,·)$, $\chi_{615}(1,·)$, $\chi_{615}(389,·)$, $\chi_{615}(449,·)$, $\chi_{615}(74,·)$, $\chi_{615}(524,·)$, $\chi_{615}(271,·)$, $\chi_{615}(16,·)$, $\chi_{615}(406,·)$, $\chi_{615}(346,·)$, $\chi_{615}(286,·)$, $\chi_{615}(31,·)$, $\chi_{615}(419,·)$, $\chi_{615}(554,·)$, $\chi_{615}(556,·)$, $\chi_{615}(494,·)$, $\chi_{615}(496,·)$, $\chi_{615}(374,·)$, $\chi_{615}(569,·)$, $\chi_{615}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{219} a^{16} - \frac{14}{219} a^{15} + \frac{14}{219} a^{14} + \frac{1}{219} a^{13} + \frac{68}{219} a^{12} - \frac{35}{219} a^{11} + \frac{47}{219} a^{10} - \frac{29}{219} a^{9} + \frac{25}{73} a^{8} - \frac{4}{219} a^{7} - \frac{47}{219} a^{6} + \frac{95}{219} a^{5} + \frac{55}{219} a^{4} - \frac{34}{219} a^{3} + \frac{76}{219} a^{2} - \frac{31}{219} a - \frac{4}{219}$, $\frac{1}{219} a^{17} - \frac{12}{73} a^{15} - \frac{22}{219} a^{14} - \frac{64}{219} a^{13} - \frac{35}{73} a^{12} - \frac{5}{219} a^{11} - \frac{101}{219} a^{10} + \frac{34}{219} a^{9} - \frac{49}{219} a^{8} - \frac{103}{219} a^{7} - \frac{52}{219} a^{6} - \frac{25}{73} a^{5} + \frac{79}{219} a^{4} - \frac{35}{219} a^{3} + \frac{28}{73} a^{2} + \frac{17}{219}$, $\frac{1}{219} a^{18} - \frac{5}{73} a^{15} + \frac{2}{219} a^{14} + \frac{77}{219} a^{13} - \frac{13}{73} a^{12} - \frac{47}{219} a^{11} + \frac{47}{219} a^{10} + \frac{25}{73} a^{9} - \frac{31}{219} a^{8} + \frac{23}{219} a^{7} - \frac{88}{219} a^{6} - \frac{26}{73} a^{5} - \frac{26}{219} a^{4} + \frac{28}{219} a^{3} - \frac{38}{219} a^{2} - \frac{4}{219} a + \frac{2}{219}$, $\frac{1}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{19} + \frac{496695074975571263679152258434303696434298746450085126415932028561067018976666265005736}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{18} - \frac{64086784438264575843284218062869232300091729218820636256736173280794719761351423654704}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{17} + \frac{479558625916305186609251322549803930416637321886465253275743677810755824838084611263630}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{16} - \frac{10091867532067255075425917153085626852035188039187219691614342980127745166330624555937311}{87545662073795449341924655342438027387444421661281316161599950708157574746053438120153607} a^{15} - \frac{18723807623478961789924021153526058360259758440147427371132740495274625660423560608144528}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{14} + \frac{118311907598553055860216383237630934045338545662193534330571495382190214185904178966884108}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{13} - \frac{9469911705101006473357049380502405636305021978003440893956675481140977410604577848343104}{87545662073795449341924655342438027387444421661281316161599950708157574746053438120153607} a^{12} + \frac{9076147512475452439225982067725171190312445946292469612858749412685136042570437815670795}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{11} - \frac{6617091404580508652663297624395070340286139867495320724798044139291885625138344635055470}{87545662073795449341924655342438027387444421661281316161599950708157574746053438120153607} a^{10} - \frac{35298096868476448476189781005849790773865114000098712584182472623950001998109772571175957}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{9} + \frac{32925995255027659456923635927501341111268089160495595724813459982013160251395455263271580}{87545662073795449341924655342438027387444421661281316161599950708157574746053438120153607} a^{8} - \frac{125278455192030965721023668058281131602790326100631428178762939429422048787903757703116542}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{7} - \frac{73091465972193916997670030340036009591283901027375739845408552091000750935301289165162314}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{6} - \frac{26801865051281253332524745626281324718003596905419423766201416223516679636546674982375313}{87545662073795449341924655342438027387444421661281316161599950708157574746053438120153607} a^{5} + \frac{44658779021500186993394016333418696686408059116469761585348065654313043264009747517306766}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{4} + \frac{100111378741111974913343822113955784809941835291689875083775111959652003051716124132872462}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{3} + \frac{55741419662581820119627526112650387885260117166741997958350283229114466377977295958576374}{262636986221386348025773966027314082162333264983843948484799852124472724238160314360460821} a^{2} - \frac{42053112601684383443987008693112779664107386810758554307167661728926481698034263712215961}{87545662073795449341924655342438027387444421661281316161599950708157574746053438120153607} a + \frac{509608285000009526550114843045716872864770915930893936449577442979559961969296691859580}{1054767012937294570384634401716120811896920742907003809175903020580211743928354676146429}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{32644}$, which has order $2089216$ (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:  \( 5104264.636551031 \) (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

\(\Q(\sqrt{41}) \), 4.0.15507225.1, 5.5.2825761.1, 10.10.327381934393961.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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$$[\ ]_{2}^{2}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41Data not computed