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Properties

Label 20.0.34425542535...000.17
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $94.81$
Ramified primes $2, 3, 5, 11$
Class number $145640$ (GRH)
Class group $[2, 72820]$ (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![1001595904, 0, 1901621760, 0, 1450009344, 0, 599886720, 0, 150756736, 0, 24136256, 0, 2490240, 0, 162528, 0, 6340, 0, 130, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 130*x^18 + 6340*x^16 + 162528*x^14 + 2490240*x^12 + 24136256*x^10 + 150756736*x^8 + 599886720*x^6 + 1450009344*x^4 + 1901621760*x^2 + 1001595904)
 
gp: K = bnfinit(x^20 + 130*x^18 + 6340*x^16 + 162528*x^14 + 2490240*x^12 + 24136256*x^10 + 150756736*x^8 + 599886720*x^6 + 1450009344*x^4 + 1901621760*x^2 + 1001595904, 1)
 

Normalized defining polynomial

\( x^{20} + 130 x^{18} + 6340 x^{16} + 162528 x^{14} + 2490240 x^{12} + 24136256 x^{10} + 150756736 x^{8} + 599886720 x^{6} + 1450009344 x^{4} + 1901621760 x^{2} + 1001595904 \)

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:  \(3442554253548220860035959357440000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.81$
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:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(1289,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(1261,·)$, $\chi_{1320}(1109,·)$, $\chi_{1320}(569,·)$, $\chi_{1320}(541,·)$, $\chi_{1320}(329,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(809,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(689,·)$, $\chi_{1320}(1141,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(901,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{2944} a^{15} + \frac{3}{1472} a^{13} - \frac{5}{368} a^{11} + \frac{11}{368} a^{9} - \frac{1}{92} a^{7} + \frac{2}{23} a^{5} + \frac{5}{46} a^{3} + \frac{7}{23} a$, $\frac{1}{5888} a^{16} + \frac{3}{2944} a^{14} - \frac{5}{736} a^{12} + \frac{11}{736} a^{10} - \frac{1}{184} a^{8} + \frac{1}{23} a^{6} + \frac{5}{92} a^{4} + \frac{7}{46} a^{2}$, $\frac{1}{253184} a^{17} + \frac{21}{126592} a^{15} + \frac{137}{31648} a^{13} + \frac{11}{1978} a^{11} + \frac{449}{15824} a^{9} + \frac{363}{7912} a^{7} + \frac{201}{1978} a^{5} - \frac{271}{1978} a^{3} + \frac{195}{989} a$, $\frac{1}{15565212171382050304} a^{18} - \frac{45696076370135}{3891303042845512576} a^{16} + \frac{1651406053490223}{972825760711378144} a^{14} + \frac{14048158681896141}{1945651521422756288} a^{12} - \frac{842765969384213}{60801610044461134} a^{10} + \frac{1236093743153805}{60801610044461134} a^{8} - \frac{3285819772110685}{243206440177844536} a^{6} - \frac{2722750037316641}{30400805022230567} a^{4} - \frac{6071027277961743}{30400805022230567} a^{2} + \frac{3717939243031}{30738933288403}$, $\frac{1}{15565212171382050304} a^{19} + \frac{824647124939}{7782606085691025152} a^{17} + \frac{611532222722307}{3891303042845512576} a^{15} - \frac{7438588771245541}{972825760711378144} a^{13} - \frac{9150065916482585}{972825760711378144} a^{11} - \frac{2771097282773933}{243206440177844536} a^{9} - \frac{3687860472171251}{60801610044461134} a^{7} - \frac{6864199888485771}{121603220088922268} a^{5} + \frac{7208191902628353}{30400805022230567} a^{3} - \frac{3454390611551837}{30400805022230567} a$

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

Class group and class number

$C_{2}\times C_{72820}$, which has order $145640$ (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:  \( 5362955.97373 \) (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

\(\Q(\sqrt{165}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-22}, \sqrt{-30})\), \(\Q(\zeta_{11})^+\), 10.10.1790566527853125.1, 10.0.5333934907699200000.1, 10.0.77265229938688.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
$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}$