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Properties

Label 20.0.35251755556...000.14
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $134.08$
Ramified primes $2, 3, 5, 11$
Class number $8577952$ (GRH)
Class group $[2, 4, 1072244]$ (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![328492898449, 0, 147156696660, 0, 49164416095, 0, 10980160088, 0, 1603780627, 0, 154173744, 0, 9791644, 0, 405952, 0, 10553, 0, 156, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 156*x^18 + 10553*x^16 + 405952*x^14 + 9791644*x^12 + 154173744*x^10 + 1603780627*x^8 + 10980160088*x^6 + 49164416095*x^4 + 147156696660*x^2 + 328492898449)
 
gp: K = bnfinit(x^20 + 156*x^18 + 10553*x^16 + 405952*x^14 + 9791644*x^12 + 154173744*x^10 + 1603780627*x^8 + 10980160088*x^6 + 49164416095*x^4 + 147156696660*x^2 + 328492898449, 1)
 

Normalized defining polynomial

\( x^{20} + 156 x^{18} + 10553 x^{16} + 405952 x^{14} + 9791644 x^{12} + 154173744 x^{10} + 1603780627 x^{8} + 10980160088 x^{6} + 49164416095 x^{4} + 147156696660 x^{2} + 328492898449 \)

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:  \(3525175555633378160676822382018560000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.08$
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$$\chi_{1320}(1,·)$, $\chi_{1320}(899,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(661,·)$, $\chi_{1320}(479,·)$, $\chi_{1320}(359,·)$, $\chi_{1320}(421,·)$, $\chi_{1320}(1319,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(299,·)$, $\chi_{1320}(301,·)$, $\chi_{1320}(239,·)$, $\chi_{1320}(1139,·)$, $\chi_{1320}(181,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(1019,·)$, $\chi_{1320}(1021,·)$, $\chi_{1320}(959,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{573143} a^{11} + \frac{77}{573143} a^{9} + \frac{2156}{573143} a^{7} + \frac{26411}{573143} a^{5} + \frac{132055}{573143} a^{3} + \frac{184877}{573143} a$, $\frac{1}{85325517839} a^{12} + \frac{30726196307}{85325517839} a^{10} + \frac{17695792281}{85325517839} a^{8} - \frac{35743463641}{85325517839} a^{6} - \frac{16132697109}{85325517839} a^{4} + \frac{28860800642}{85325517839} a^{2} - \frac{61125}{148873}$, $\frac{1}{85325517839} a^{13} + \frac{91}{85325517839} a^{11} + \frac{40893183141}{85325517839} a^{9} + \frac{16504855566}{85325517839} a^{7} + \frac{5299208844}{85325517839} a^{5} - \frac{34630705271}{85325517839} a^{3} + \frac{29664457957}{85325517839} a$, $\frac{1}{85325517839} a^{14} - \frac{24774109948}{85325517839} a^{10} + \frac{27372596936}{85325517839} a^{8} + \frac{15584722293}{85325517839} a^{6} - \frac{17089071615}{85325517839} a^{4} - \frac{36902865295}{85325517839} a^{2} + \frac{54074}{148873}$, $\frac{1}{85325517839} a^{15} - \frac{5145}{85325517839} a^{11} - \frac{27508243530}{85325517839} a^{9} + \frac{14780510347}{85325517839} a^{7} + \frac{15722090966}{85325517839} a^{5} + \frac{41827429771}{85325517839} a^{3} + \frac{4693719132}{85325517839} a$, $\frac{1}{85325517839} a^{16} + \frac{35912718157}{85325517839} a^{10} + \frac{17304261879}{85325517839} a^{8} - \frac{7907398934}{85325517839} a^{6} - \frac{24495856526}{85325517839} a^{4} + \frac{27111982362}{85325517839} a^{2} - \frac{68349}{148873}$, $\frac{1}{85325517839} a^{17} - \frac{64506}{85325517839} a^{11} - \frac{17563432324}{85325517839} a^{9} + \frac{39703377450}{85325517839} a^{7} - \frac{38542470695}{85325517839} a^{5} + \frac{42204429356}{85325517839} a^{3} + \frac{33150985588}{85325517839} a$, $\frac{1}{85325517839} a^{18} - \frac{19998335113}{85325517839} a^{10} + \frac{39702605494}{85325517839} a^{8} - \frac{40265051583}{85325517839} a^{6} + \frac{16460280646}{85325517839} a^{4} + \frac{10483469299}{85325517839} a^{2} - \frac{27845}{148873}$, $\frac{1}{85325517839} a^{19} + \frac{72723}{85325517839} a^{11} - \frac{41604830075}{85325517839} a^{9} - \frac{13084265862}{85325517839} a^{7} + \frac{29454213832}{85325517839} a^{5} - \frac{9872382610}{85325517839} a^{3} - \frac{10327152372}{85325517839} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{1072244}$, which has order $8577952$ (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:  \( 530208.2507325789 \) (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

\(\Q(\sqrt{-330}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-165})\), \(\Q(\zeta_{11})^+\), 10.0.58673283984691200000.1, 10.0.1833540124521600000.1, 10.10.7024111812608.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed