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Properties

Label 20.0.73790913002...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{32}\cdot 13^{15}$
Root discriminant $311.46$
Ramified primes $2, 3, 5, 13$
Class number $1258365128$ (GRH)
Class group $[2, 2, 314591282]$ (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![926878489057, 156576885230, 525018844775, 135865272390, 131515368985, 26884464018, 20216277025, 1136898640, 1066176320, -26142000, 113584520, -1690500, 9780190, -23640, 437600, -964, 10975, -20, 155, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 155*x^18 - 20*x^17 + 10975*x^16 - 964*x^15 + 437600*x^14 - 23640*x^13 + 9780190*x^12 - 1690500*x^11 + 113584520*x^10 - 26142000*x^9 + 1066176320*x^8 + 1136898640*x^7 + 20216277025*x^6 + 26884464018*x^5 + 131515368985*x^4 + 135865272390*x^3 + 525018844775*x^2 + 156576885230*x + 926878489057)
 
gp: K = bnfinit(x^20 + 155*x^18 - 20*x^17 + 10975*x^16 - 964*x^15 + 437600*x^14 - 23640*x^13 + 9780190*x^12 - 1690500*x^11 + 113584520*x^10 - 26142000*x^9 + 1066176320*x^8 + 1136898640*x^7 + 20216277025*x^6 + 26884464018*x^5 + 131515368985*x^4 + 135865272390*x^3 + 525018844775*x^2 + 156576885230*x + 926878489057, 1)
 

Normalized defining polynomial

\( x^{20} + 155 x^{18} - 20 x^{17} + 10975 x^{16} - 964 x^{15} + 437600 x^{14} - 23640 x^{13} + 9780190 x^{12} - 1690500 x^{11} + 113584520 x^{10} - 26142000 x^{9} + 1066176320 x^{8} + 1136898640 x^{7} + 20216277025 x^{6} + 26884464018 x^{5} + 131515368985 x^{4} + 135865272390 x^{3} + 525018844775 x^{2} + 156576885230 x + 926878489057 \)

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:  \(73790913002662234133129882812500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{32}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $311.46$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3900=2^{2}\cdot 3\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3900}(1,·)$, $\chi_{3900}(3011,·)$, $\chi_{3900}(961,·)$, $\chi_{3900}(2891,·)$, $\chi_{3900}(781,·)$, $\chi_{3900}(1741,·)$, $\chi_{3900}(3671,·)$, $\chi_{3900}(2521,·)$, $\chi_{3900}(3791,·)$, $\chi_{3900}(2341,·)$, $\chi_{3900}(1561,·)$, $\chi_{3900}(3301,·)$, $\chi_{3900}(551,·)$, $\chi_{3900}(1451,·)$, $\chi_{3900}(3121,·)$, $\chi_{3900}(1331,·)$, $\chi_{3900}(181,·)$, $\chi_{3900}(2231,·)$, $\chi_{3900}(671,·)$, $\chi_{3900}(2111,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{4}{9} a^{9} - \frac{2}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{14} - \frac{2}{9} a^{9} - \frac{4}{9} a^{8} + \frac{4}{9} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{4}{9} a^{3} + \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{15} - \frac{4}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{6} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{63} a^{16} + \frac{2}{63} a^{15} + \frac{1}{21} a^{14} - \frac{1}{21} a^{13} - \frac{2}{63} a^{11} - \frac{16}{63} a^{9} - \frac{3}{7} a^{8} + \frac{4}{21} a^{7} - \frac{8}{63} a^{6} + \frac{13}{63} a^{5} + \frac{22}{63} a^{4} + \frac{26}{63} a^{3} + \frac{1}{21} a^{2} + \frac{13}{63} a + \frac{8}{63}$, $\frac{1}{2709} a^{17} - \frac{1}{903} a^{16} - \frac{1}{43} a^{15} - \frac{46}{2709} a^{14} - \frac{20}{2709} a^{13} + \frac{110}{2709} a^{12} + \frac{136}{2709} a^{11} - \frac{1}{301} a^{10} + \frac{424}{2709} a^{9} - \frac{73}{387} a^{8} - \frac{67}{903} a^{7} + \frac{179}{2709} a^{6} - \frac{1352}{2709} a^{5} - \frac{19}{387} a^{4} - \frac{127}{2709} a^{3} + \frac{1132}{2709} a^{2} - \frac{260}{2709} a - \frac{5}{63}$, $\frac{1}{24013918686145966497146897041046787} a^{18} + \frac{2156031597066441657054694325201}{24013918686145966497146897041046787} a^{17} + \frac{20074711331479939683180682379877}{2668213187349551833016321893449643} a^{16} - \frac{3454204010151998597404797401291}{79780460751315503312780388840687} a^{15} - \frac{1224101666227468607482199021528644}{24013918686145966497146897041046787} a^{14} + \frac{1199772314411883756512762294883469}{24013918686145966497146897041046787} a^{13} - \frac{649952052506907583501353280515763}{24013918686145966497146897041046787} a^{12} - \frac{196605634145049475770442469098705}{24013918686145966497146897041046787} a^{11} + \frac{1247651115523458366456420950749781}{24013918686145966497146897041046787} a^{10} - \frac{1625646038981196336091755303034795}{24013918686145966497146897041046787} a^{9} - \frac{7809738511780212783734161337036668}{24013918686145966497146897041046787} a^{8} + \frac{4091833657424548125856718097040981}{24013918686145966497146897041046787} a^{7} + \frac{4711169049601067050617913343070565}{24013918686145966497146897041046787} a^{6} + \frac{623972904014149658742696089806154}{8004639562048655499048965680348929} a^{5} + \frac{11612374302367816187933683910284259}{24013918686145966497146897041046787} a^{4} - \frac{11804018921877889196198961116599717}{24013918686145966497146897041046787} a^{3} + \frac{4201313977633296149475861543076871}{24013918686145966497146897041046787} a^{2} - \frac{5383590902991505291078822372183289}{24013918686145966497146897041046787} a - \frac{119059287770762613868798940034212}{558463225259208523189462721884809}$, $\frac{1}{76251313053892411628337909983846342071906760930301336572953846582941} a^{19} + \frac{161183431607093266956671254138378}{10893044721984630232619701426263763153129537275757333796136263797563} a^{18} - \frac{7239469373340608044475510250812567671734296577562122307899492244}{76251313053892411628337909983846342071906760930301336572953846582941} a^{17} - \frac{93815524192028719386035553267194894440693627544934171703008662084}{76251313053892411628337909983846342071906760930301336572953846582941} a^{16} + \frac{1789833106498750222237918362887314803422188230413822220912233352193}{76251313053892411628337909983846342071906760930301336572953846582941} a^{15} - \frac{1844034645877513553864132344223496299733218758378854299380882409}{161892384403168602183307664509227902488124757813803262362959334571} a^{14} - \frac{663339891593890375040187077729629540857793320073165176905494628669}{76251313053892411628337909983846342071906760930301336572953846582941} a^{13} - \frac{48592033764608714895552307012163374248712513206319964558206546655}{25417104351297470542779303327948780690635586976767112190984615527647} a^{12} - \frac{177717356335649916829384126266090070560632364437270745889259685370}{8472368117099156847593101109316260230211862325589037396994871842549} a^{11} - \frac{322009367074835991635260963396088136469764014958281564853958680480}{8472368117099156847593101109316260230211862325589037396994871842549} a^{10} - \frac{19214322145246823667203101906599883965573414091524936989238602722654}{76251313053892411628337909983846342071906760930301336572953846582941} a^{9} - \frac{32146438622594716216674183104990202738650286363544528135690120414775}{76251313053892411628337909983846342071906760930301336572953846582941} a^{8} + \frac{6626329214517768865013275230953466212503182554922103052488731695626}{76251313053892411628337909983846342071906760930301336572953846582941} a^{7} + \frac{2240543986498039232493133625281474788466494289879167196384423846353}{8472368117099156847593101109316260230211862325589037396994871842549} a^{6} - \frac{11912705242587674563033452057008430726119842972724185952382729853349}{25417104351297470542779303327948780690635586976767112190984615527647} a^{5} + \frac{4911326955422540783071381877775983825531234254105183182459203258317}{25417104351297470542779303327948780690635586976767112190984615527647} a^{4} + \frac{12123889043543581917117682922518769542861716272430566233025595367813}{76251313053892411628337909983846342071906760930301336572953846582941} a^{3} + \frac{832261132294515094852947818407887369496756404917316729184263159115}{10893044721984630232619701426263763153129537275757333796136263797563} a^{2} + \frac{10534735506735273156665786528476899882343830181406608877723917483238}{25417104351297470542779303327948780690635586976767112190984615527647} a - \frac{9923888380688378992549525570031740813762011250639474590374765}{21894856837062404472830575772802301625792690985274946304098057}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{314591282}$, which has order $1258365128$ (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:  \( 47804543.59281621 \) (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

\(\Q(\sqrt{13}) \), 4.0.316368.2, 5.5.390625.1, 10.10.56654815673828125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
13Data not computed