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Properties

Label 20.0.16082319642...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $102.40$
Ramified primes $2, 5, 7, 11$
Class number $1214840$ (GRH)
Class group $[2, 22, 27610]$ (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![26717460001, -16042910010, 20850916755, -10547164366, 7729934861, -3320524362, 1768870593, -651765860, 275855648, -87461308, 30565866, -8305814, 2430720, -557978, 136388, -25620, 5115, -732, 113, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 113*x^18 - 732*x^17 + 5115*x^16 - 25620*x^15 + 136388*x^14 - 557978*x^13 + 2430720*x^12 - 8305814*x^11 + 30565866*x^10 - 87461308*x^9 + 275855648*x^8 - 651765860*x^7 + 1768870593*x^6 - 3320524362*x^5 + 7729934861*x^4 - 10547164366*x^3 + 20850916755*x^2 - 16042910010*x + 26717460001)
 
gp: K = bnfinit(x^20 - 10*x^19 + 113*x^18 - 732*x^17 + 5115*x^16 - 25620*x^15 + 136388*x^14 - 557978*x^13 + 2430720*x^12 - 8305814*x^11 + 30565866*x^10 - 87461308*x^9 + 275855648*x^8 - 651765860*x^7 + 1768870593*x^6 - 3320524362*x^5 + 7729934861*x^4 - 10547164366*x^3 + 20850916755*x^2 - 16042910010*x + 26717460001, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 113 x^{18} - 732 x^{17} + 5115 x^{16} - 25620 x^{15} + 136388 x^{14} - 557978 x^{13} + 2430720 x^{12} - 8305814 x^{11} + 30565866 x^{10} - 87461308 x^{9} + 275855648 x^{8} - 651765860 x^{7} + 1768870593 x^{6} - 3320524362 x^{5} + 7729934861 x^{4} - 10547164366 x^{3} + 20850916755 x^{2} - 16042910010 x + 26717460001 \)

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:  \(16082319642092842466929804994560000000000=2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.40$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1540=2^{2}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1540}(1,·)$, $\chi_{1540}(1539,·)$, $\chi_{1540}(69,·)$, $\chi_{1540}(841,·)$, $\chi_{1540}(139,·)$, $\chi_{1540}(141,·)$, $\chi_{1540}(211,·)$, $\chi_{1540}(1049,·)$, $\chi_{1540}(1051,·)$, $\chi_{1540}(1119,·)$, $\chi_{1540}(489,·)$, $\chi_{1540}(1189,·)$, $\chi_{1540}(421,·)$, $\chi_{1540}(491,·)$, $\chi_{1540}(1329,·)$, $\chi_{1540}(351,·)$, $\chi_{1540}(1399,·)$, $\chi_{1540}(1401,·)$, $\chi_{1540}(699,·)$, $\chi_{1540}(1471,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{71765640719647646911384587069641} a^{18} - \frac{9}{71765640719647646911384587069641} a^{17} - \frac{34096356976956076690238079781094}{71765640719647646911384587069641} a^{16} - \frac{14291707062941974123633710029608}{71765640719647646911384587069641} a^{15} + \frac{25703054800371392073302359250875}{71765640719647646911384587069641} a^{14} - \frac{1582150720762844260911176308340}{71765640719647646911384587069641} a^{13} - \frac{914666020533136589547699261801}{3120245248680332474408025524767} a^{12} - \frac{1121316122836469496213906720067}{71765640719647646911384587069641} a^{11} - \frac{3475849877168649585518220843288}{71765640719647646911384587069641} a^{10} - \frac{17369290484020571563813332837166}{71765640719647646911384587069641} a^{9} + \frac{4480153353277637540686698802480}{71765640719647646911384587069641} a^{8} + \frac{25675149052571946542257640874947}{71765640719647646911384587069641} a^{7} + \frac{2288819577038486868077421244681}{71765640719647646911384587069641} a^{6} + \frac{19162721648534179134768339084262}{71765640719647646911384587069641} a^{5} - \frac{4113015350660634284444289355081}{71765640719647646911384587069641} a^{4} + \frac{25931489080713855310874432804428}{71765640719647646911384587069641} a^{3} - \frac{8389990073704724605341858449511}{71765640719647646911384587069641} a^{2} + \frac{2235607628806588699744765283913}{71765640719647646911384587069641} a - \frac{23381279295595903996007970951679}{71765640719647646911384587069641}$, $\frac{1}{11143355249090700722305942193567746278199} a^{19} + \frac{77637110}{11143355249090700722305942193567746278199} a^{18} + \frac{258778900352572556917568967045182200684}{11143355249090700722305942193567746278199} a^{17} + \frac{31518001322342867613150286672980716311}{484493706482204379230693138850771577313} a^{16} - \frac{4610857748480288147592145131251285742840}{11143355249090700722305942193567746278199} a^{15} - \frac{184204076783227000802919249691522235300}{11143355249090700722305942193567746278199} a^{14} + \frac{1064752789684496435882251909901211331037}{11143355249090700722305942193567746278199} a^{13} + \frac{767712180622798440048827757894119347084}{11143355249090700722305942193567746278199} a^{12} + \frac{691628787776167677755917296234316172960}{11143355249090700722305942193567746278199} a^{11} - \frac{17340689454704163826951444025541869623}{484493706482204379230693138850771577313} a^{10} + \frac{3177132213684189600026273472149915734616}{11143355249090700722305942193567746278199} a^{9} + \frac{1670038470524183154834217181227028933656}{11143355249090700722305942193567746278199} a^{8} + \frac{5207779036657858419677486067558507855330}{11143355249090700722305942193567746278199} a^{7} - \frac{5167689223904393037883008040296344022624}{11143355249090700722305942193567746278199} a^{6} - \frac{253097586749593917581434720837919606179}{11143355249090700722305942193567746278199} a^{5} - \frac{2576292902041542817545016113096888113027}{11143355249090700722305942193567746278199} a^{4} + \frac{4513099636584929519685061450349043631432}{11143355249090700722305942193567746278199} a^{3} + \frac{819045120885531804732059855775754896253}{11143355249090700722305942193567746278199} a^{2} + \frac{139375347790789718939429879704758572270}{484493706482204379230693138850771577313} a + \frac{4873331688876877494635043171308845701445}{11143355249090700722305942193567746278199}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{27610}$, which has order $1214840$ (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:  \( 281202.4907663525 \) (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

\(\Q(\sqrt{-385}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{11}, \sqrt{-35})\), \(\Q(\zeta_{11})^+\), 10.0.126816085896438400000.1, \(\Q(\zeta_{44})^+\), 10.0.11258530353021875.4

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$