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Properties

Label 20.0.17374741604...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{35}\cdot 11^{18}$
Root discriminant $409.26$
Ramified primes $2, 5, 11$
Class number $1477684480$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 23088820]$ (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![16139001528320, 0, 116430854144000, 0, 44882044064000, 0, 6572678182400, 0, 486238403200, 0, 20634408960, 0, 532932400, 0, 8518400, 0, 82280, 0, 440, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 440*x^18 + 82280*x^16 + 8518400*x^14 + 532932400*x^12 + 20634408960*x^10 + 486238403200*x^8 + 6572678182400*x^6 + 44882044064000*x^4 + 116430854144000*x^2 + 16139001528320)
 
gp: K = bnfinit(x^20 + 440*x^18 + 82280*x^16 + 8518400*x^14 + 532932400*x^12 + 20634408960*x^10 + 486238403200*x^8 + 6572678182400*x^6 + 44882044064000*x^4 + 116430854144000*x^2 + 16139001528320, 1)
 

Normalized defining polynomial

\( x^{20} + 440 x^{18} + 82280 x^{16} + 8518400 x^{14} + 532932400 x^{12} + 20634408960 x^{10} + 486238403200 x^{8} + 6572678182400 x^{6} + 44882044064000 x^{4} + 116430854144000 x^{2} + 16139001528320 \)

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:  \(17374741604663223378125000000000000000000000000000000=2^{30}\cdot 5^{35}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $409.26$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(1267,·)$, $\chi_{2200}(961,·)$, $\chi_{2200}(1609,·)$, $\chi_{2200}(1163,·)$, $\chi_{2200}(987,·)$, $\chi_{2200}(1681,·)$, $\chi_{2200}(1849,·)$, $\chi_{2200}(1883,·)$, $\chi_{2200}(929,·)$, $\chi_{2200}(227,·)$, $\chi_{2200}(1489,·)$, $\chi_{2200}(1769,·)$, $\chi_{2200}(43,·)$, $\chi_{2200}(1403,·)$, $\chi_{2200}(307,·)$, $\chi_{2200}(641,·)$, $\chi_{2200}(1721,·)$, $\chi_{2200}(347,·)$, $\chi_{2200}(1723,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{16} a^{5} - \frac{1}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{1216} a^{6} - \frac{37}{608} a^{4} + \frac{53}{304} a^{2} - \frac{9}{19}$, $\frac{1}{1216} a^{7} + \frac{1}{608} a^{5} + \frac{15}{304} a^{3} - \frac{17}{76} a$, $\frac{1}{2432} a^{8} + \frac{13}{152} a^{4} + \frac{65}{304} a^{2} + \frac{9}{19}$, $\frac{1}{2432} a^{9} + \frac{7}{304} a^{5} - \frac{49}{304} a^{3} + \frac{17}{76} a$, $\frac{1}{8132608} a^{10} + \frac{5}{184832} a^{8} - \frac{71}{184832} a^{6} - \frac{91}{46208} a^{4} + \frac{9107}{46208} a^{2} + \frac{207}{1444}$, $\frac{1}{8132608} a^{11} + \frac{5}{184832} a^{9} - \frac{71}{184832} a^{7} - \frac{91}{46208} a^{5} + \frac{9107}{46208} a^{3} + \frac{207}{1444} a$, $\frac{1}{16265216} a^{12} + \frac{45}{369664} a^{8} + \frac{7}{46208} a^{6} + \frac{205}{4864} a^{4} - \frac{2645}{23104} a^{2} - \frac{213}{722}$, $\frac{1}{16265216} a^{13} + \frac{45}{369664} a^{9} + \frac{7}{46208} a^{7} - \frac{99}{4864} a^{5} + \frac{243}{23104} a^{3} + \frac{657}{1444} a$, $\frac{1}{357834752} a^{14} + \frac{25}{184832} a^{8} - \frac{1}{11552} a^{6} - \frac{7525}{63536} a^{4} + \frac{4249}{46208} a^{2} + \frac{565}{1444}$, $\frac{1}{41508831232} a^{15} + \frac{15}{1886765056} a^{13} - \frac{27}{471691264} a^{11} - \frac{7415}{42881024} a^{9} + \frac{2575}{10720256} a^{7} + \frac{3028031}{117922816} a^{5} + \frac{1297857}{5360128} a^{3} - \frac{48087}{167504} a$, $\frac{1}{3154671173632} a^{16} + \frac{131}{143394144256} a^{14} + \frac{611}{35848536064} a^{12} - \frac{107}{3258957824} a^{10} - \frac{116731}{814739456} a^{8} - \frac{104359}{471691264} a^{6} - \frac{40754579}{407369728} a^{4} + \frac{1011}{8816} a^{2} - \frac{5973}{13718}$, $\frac{1}{34701382909952} a^{17} + \frac{17}{1577335586816} a^{15} + \frac{1}{17924268032} a^{13} + \frac{1641}{35848536064} a^{11} + \frac{15075}{407369728} a^{9} + \frac{763101}{5188603904} a^{7} + \frac{57380079}{4481067008} a^{5} + \frac{51675}{282112} a^{3} - \frac{931253}{3182576} a$, $\frac{1}{69402765819904} a^{18} + \frac{105}{83017662464} a^{14} - \frac{79}{3773530112} a^{12} + \frac{69}{35848536064} a^{10} + \frac{5098429}{197166948352} a^{8} - \frac{247669}{2240533504} a^{6} + \frac{285709811}{4481067008} a^{4} - \frac{3966673}{25460608} a^{2} + \frac{2385}{27436}$, $\frac{1}{69402765819904} a^{19} - \frac{1}{83017662464} a^{15} - \frac{45}{3773530112} a^{13} - \frac{125}{3258957824} a^{11} - \frac{17896169}{197166948352} a^{9} + \frac{324575}{1120266752} a^{7} + \frac{100415465}{4481067008} a^{5} + \frac{14047345}{101842432} a^{3} - \frac{517255}{3182576} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{23088820}$, which has order $1477684480$ (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

$C_{20}$ (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

\(\Q(\sqrt{5}) \), 4.0.968000.5, 5.5.5719140625.4, 10.10.163542847442626953125.4

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 $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
5Data not computed
$11$11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$