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Properties

Label 20.0.20038303066...000.39
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $462.48$
Ramified primes $2, 3, 5, 11$
Class number $8080609280$ (GRH)
Class group $[2, 2, 4, 4, 4, 4, 7891220]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![165438348587866849, -11931706889206280, 14446541807915030, -113622665835360, 611194037644355, 4633838667656, 20414578791200, 110410123560, 472877380525, 5112051560, 8915966070, 4631880, 140041165, 842160, 1650440, -2552, 18755, 0, 110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 110*x^18 + 18755*x^16 - 2552*x^15 + 1650440*x^14 + 842160*x^13 + 140041165*x^12 + 4631880*x^11 + 8915966070*x^10 + 5112051560*x^9 + 472877380525*x^8 + 110410123560*x^7 + 20414578791200*x^6 + 4633838667656*x^5 + 611194037644355*x^4 - 113622665835360*x^3 + 14446541807915030*x^2 - 11931706889206280*x + 165438348587866849)
 
gp: K = bnfinit(x^20 + 110*x^18 + 18755*x^16 - 2552*x^15 + 1650440*x^14 + 842160*x^13 + 140041165*x^12 + 4631880*x^11 + 8915966070*x^10 + 5112051560*x^9 + 472877380525*x^8 + 110410123560*x^7 + 20414578791200*x^6 + 4633838667656*x^5 + 611194037644355*x^4 - 113622665835360*x^3 + 14446541807915030*x^2 - 11931706889206280*x + 165438348587866849, 1)
 

Normalized defining polynomial

\( x^{20} + 110 x^{18} + 18755 x^{16} - 2552 x^{15} + 1650440 x^{14} + 842160 x^{13} + 140041165 x^{12} + 4631880 x^{11} + 8915966070 x^{10} + 5112051560 x^{9} + 472877380525 x^{8} + 110410123560 x^{7} + 20414578791200 x^{6} + 4633838667656 x^{5} + 611194037644355 x^{4} - 113622665835360 x^{3} + 14446541807915030 x^{2} - 11931706889206280 x + 165438348587866849 \)

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:  \(200383030666749741651348266601562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.48$
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$$\chi_{3300}(1,·)$, $\chi_{3300}(2821,·)$, $\chi_{3300}(961,·)$, $\chi_{3300}(1609,·)$, $\chi_{3300}(1741,·)$, $\chi_{3300}(1681,·)$, $\chi_{3300}(3299,·)$, $\chi_{3300}(1559,·)$, $\chi_{3300}(1691,·)$, $\chi_{3300}(479,·)$, $\chi_{3300}(2339,·)$, $\chi_{3300}(1489,·)$, $\chi_{3300}(1451,·)$, $\chi_{3300}(2029,·)$, $\chi_{3300}(431,·)$, $\chi_{3300}(1619,·)$, $\chi_{3300}(1811,·)$, $\chi_{3300}(2869,·)$, $\chi_{3300}(1271,·)$, $\chi_{3300}(1849,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{44} a^{6} - \frac{1}{22} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{44} a^{7} - \frac{1}{22} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{44} a^{8} - \frac{1}{22} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{484} a^{9} - \frac{1}{22} a^{5} + \frac{4}{11} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{968} a^{10} - \frac{1}{88} a^{8} - \frac{1}{88} a^{6} - \frac{1}{22} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{968} a^{11} - \frac{1}{968} a^{9} - \frac{1}{88} a^{7} - \frac{1}{88} a^{5} - \frac{2}{11} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a$, $\frac{1}{36784} a^{12} + \frac{1}{9196} a^{11} + \frac{1}{2299} a^{10} - \frac{1}{2299} a^{9} - \frac{7}{836} a^{8} + \frac{2}{209} a^{7} + \frac{5}{1672} a^{6} - \frac{31}{836} a^{5} + \frac{1}{418} a^{4} - \frac{13}{38} a^{3} + \frac{7}{38} a^{2} - \frac{9}{19} a + \frac{3}{16}$, $\frac{1}{404624} a^{13} + \frac{5}{18392} a^{10} - \frac{9}{9196} a^{9} - \frac{61}{18392} a^{8} + \frac{5}{1672} a^{7} + \frac{15}{1672} a^{6} - \frac{15}{418} a^{5} - \frac{145}{1672} a^{4} - \frac{94}{209} a^{3} - \frac{53}{152} a^{2} - \frac{129}{304} a - \frac{1}{8}$, $\frac{1}{809248} a^{14} - \frac{1}{809248} a^{13} - \frac{1}{73568} a^{12} - \frac{1}{2299} a^{11} - \frac{3}{9196} a^{10} - \frac{1}{2299} a^{9} + \frac{61}{36784} a^{8} + \frac{13}{3344} a^{7} + \frac{15}{3344} a^{6} - \frac{1}{836} a^{5} + \frac{369}{1672} a^{4} - \frac{19}{88} a^{3} - \frac{193}{608} a^{2} - \frac{107}{608} a - \frac{11}{32}$, $\frac{1}{8901728} a^{15} - \frac{1}{73568} a^{12} - \frac{1}{18392} a^{11} + \frac{85}{202312} a^{10} - \frac{35}{36784} a^{9} - \frac{45}{18392} a^{8} + \frac{1}{1672} a^{7} + \frac{17}{3344} a^{6} + \frac{159}{4598} a^{5} + \frac{259}{1672} a^{4} - \frac{2809}{6688} a^{3} + \frac{35}{76} a^{2} + \frac{29}{152} a + \frac{1}{32}$, $\frac{1}{17803456} a^{16} - \frac{1}{1618496} a^{14} - \frac{1}{147136} a^{12} - \frac{7}{202312} a^{11} - \frac{3}{6688} a^{10} - \frac{1}{18392} a^{9} - \frac{1}{608} a^{8} - \frac{7}{1672} a^{7} + \frac{359}{73568} a^{6} + \frac{73}{1672} a^{5} - \frac{329}{13376} a^{4} + \frac{35}{152} a^{3} + \frac{121}{1216} a^{2} - \frac{1}{152} a + \frac{5}{64}$, $\frac{1}{17803456} a^{17} - \frac{1}{17803456} a^{15} + \frac{1}{1618496} a^{13} + \frac{5}{809248} a^{12} - \frac{21}{73568} a^{11} + \frac{5}{10648} a^{10} - \frac{5}{6688} a^{9} + \frac{153}{18392} a^{8} + \frac{711}{73568} a^{7} - \frac{1}{176} a^{6} + \frac{4221}{147136} a^{5} + \frac{11}{152} a^{4} + \frac{425}{13376} a^{3} + \frac{9}{38} a^{2} + \frac{443}{1216} a - \frac{7}{32}$, $\frac{1}{11358604928} a^{18} - \frac{1}{35606912} a^{17} + \frac{3}{258150112} a^{16} - \frac{1}{35606912} a^{15} + \frac{17}{46936384} a^{14} - \frac{1}{1874048} a^{13} - \frac{257}{93872768} a^{12} - \frac{185}{1618496} a^{11} + \frac{3767}{11734096} a^{10} + \frac{49}{147136} a^{9} + \frac{79507}{11734096} a^{8} - \frac{7}{147136} a^{7} + \frac{91717}{8533888} a^{6} - \frac{10277}{294272} a^{5} - \frac{62877}{193952} a^{4} - \frac{829}{2432} a^{3} + \frac{4301}{35264} a^{2} - \frac{675}{2432} a + \frac{729}{3712}$, $\frac{1}{99205172113516020104668549793338403960260005998882832536551290669664384} a^{19} - \frac{38854613446569329660022682173882834232068868662875587515929}{3420868003914345520850639648046151860698620896513201121950044505850496} a^{18} + \frac{57658457040631376489127920436856874684984322811356073140541325}{4509326005159819095666752263333563816375454818131037842570513212257472} a^{17} + \frac{7495157548588038572564782876742984247516996069159629382690693}{310988000355849592804603604367831987336238263319381920177276773259136} a^{16} + \frac{3942123798138750499125342493175002007397720547752355609984495}{2254663002579909547833376131666781908187727409065518921285256606128736} a^{15} + \frac{182439597499616365691858023485998353698292987479603566204897231}{310988000355849592804603604367831987336238263319381920177276773259136} a^{14} + \frac{10629711637640069304310687857399031554254883283658679150414890783}{9018652010319638191333504526667127632750909636262075685141026424514944} a^{13} + \frac{9292678206191570321236527163647947399475045844494388286794675}{1285074381635742119027287621354677633620819269914801323046598236608} a^{12} - \frac{2408897595470617937197525885904820184938042574401444963417165361}{10787861256363203578150125031898478029606351239547937422417495723104} a^{11} + \frac{6936983623262201868776075354651396717718658512608508641282230497}{14135818197993163309300163834901453969829011969062814553512580602688} a^{10} - \frac{64840946511364464199512177934191293740891390104941358505497855619}{204969363870900867984852375606071082562520673551410811025932418738976} a^{9} + \frac{152450286008868271809284135703929167285901893826452636280133263241}{14135818197993163309300163834901453969829011969062814553512580602688} a^{8} - \frac{476230059398096212780633400303244235267850851101638472681906967255}{74534314134873042903582682038571302750007517655058476736702697723264} a^{7} + \frac{269943147142130749233118038596537033082831055931813386484118569}{2570148763271484238054575242709355267241638539829602646093196473216} a^{6} + \frac{131281461404726207794519726686460566517920670240622903194114600021}{37267157067436521451791341019285651375003758827529238368351348861632} a^{5} + \frac{39094002073978470326954059520872575911318253504616786157814007083}{233649887570134930732234112973577751567421685439054786008472406656} a^{4} + \frac{396683147271587425213446301568894871667563262215587186210422381091}{846980842441739123904348659529219349431903609716573599280712474128} a^{3} + \frac{8028934646629096543595750864752073512196984991596859065208538767}{21240898870012266430203101179416159233401971403550435091679309696} a^{2} - \frac{2543617888445232061469743165119008128212396818739853681677129337}{32420319327913459288204733379108874619403008984366453560984209536} a - \frac{2261900969179436332066639501584288923114553634220828926849443}{7354881880198153196053705394534681175000682618957906887700592}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{7891220}$, which has order $8080609280$ (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:  \( 4235385044.5954027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), 5.5.5719140625.4, 10.10.163542847442626953125.4, 10.0.89528326392656250000000000.4, 10.0.447641631963281250000000000.2

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
3Data not computed
5Data not computed
$11$11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$