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Properties

Label 20.0.20403517554...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 401^{8}$
Root discriminant $36.77$
Ramified primes $5, 401$
Class number $29$ (GRH)
Class group $[29]$ (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 13, -46, 174, 49, 309, 172, 837, -501, 704, -412, 538, -78, 123, -18, 30, -7, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 6 x^{18} - 7 x^{17} + 30 x^{16} - 18 x^{15} + 123 x^{14} - 78 x^{13} + 538 x^{12} - 412 x^{11} + 704 x^{10} - 501 x^{9} + 837 x^{8} + 172 x^{7} + 309 x^{6} + 49 x^{5} + 174 x^{4} - 46 x^{3} + 13 x^{2} - 3 x + 1 \)

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:  \(20403517554797011816436767578125=5^{15}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.77$
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:  $5, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1417708231} a^{17} + \frac{384167857}{1417708231} a^{16} + \frac{637500389}{1417708231} a^{15} - \frac{134369331}{1417708231} a^{14} + \frac{605418008}{1417708231} a^{13} - \frac{311749414}{1417708231} a^{12} + \frac{208049519}{1417708231} a^{11} + \frac{78541616}{1417708231} a^{10} - \frac{549468139}{1417708231} a^{9} + \frac{686496298}{1417708231} a^{8} - \frac{644161275}{1417708231} a^{7} - \frac{489623395}{1417708231} a^{6} + \frac{232383097}{1417708231} a^{5} + \frac{321375228}{1417708231} a^{4} + \frac{128034204}{1417708231} a^{3} + \frac{158908518}{1417708231} a^{2} - \frac{15990677}{1417708231} a - \frac{149797153}{1417708231}$, $\frac{1}{1417708231} a^{18} + \frac{662726879}{1417708231} a^{16} + \frac{484932519}{1417708231} a^{15} - \frac{6714589}{1417708231} a^{14} + \frac{397177271}{1417708231} a^{13} + \frac{506524719}{1417708231} a^{12} + \frac{674365493}{1417708231} a^{11} + \frac{236237116}{1417708231} a^{10} - \frac{276252692}{1417708231} a^{9} - \frac{644775711}{1417708231} a^{8} + \frac{466747914}{1417708231} a^{7} - \frac{638809667}{1417708231} a^{6} + \frac{506107805}{1417708231} a^{5} + \frac{377415341}{1417708231} a^{4} + \frac{150752643}{1417708231} a^{3} - \frac{569769128}{1417708231} a^{2} - \frac{177763377}{1417708231} a + \frac{255558026}{1417708231}$, $\frac{1}{1417708231} a^{19} - \frac{130521986}{1417708231} a^{16} - \frac{567182095}{1417708231} a^{15} - \frac{517410571}{1417708231} a^{14} - \frac{437154121}{1417708231} a^{13} - \frac{449439250}{1417708231} a^{12} - \frac{245569725}{1417708231} a^{11} - \frac{597585616}{1417708231} a^{10} - \frac{646655140}{1417708231} a^{9} - \frac{167879803}{1417708231} a^{8} + \frac{221075082}{1417708231} a^{7} - \frac{405494546}{1417708231} a^{6} - \frac{117115702}{1417708231} a^{5} + \frac{14865276}{1417708231} a^{4} + \frac{8276476}{1417708231} a^{3} - \frac{277849714}{1417708231} a^{2} + \frac{463062711}{1417708231} a + \frac{456636863}{1417708231}$

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

Class group and class number

$C_{29}$, which has order $29$ (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:  \( \frac{325947825}{1417708231} a^{19} - \frac{216420225}{1417708231} a^{18} + \frac{1847037675}{1417708231} a^{17} - \frac{1629739125}{1417708231} a^{16} + \frac{9017889825}{1417708231} a^{15} - \frac{2607582600}{1417708231} a^{14} + \frac{38154039180}{1417708231} a^{13} - \frac{12060069525}{1417708231} a^{12} + \frac{166885286400}{1417708231} a^{11} - \frac{75837193950}{1417708231} a^{10} + \frac{184703767500}{1417708231} a^{9} - \frac{88343307058}{1417708231} a^{8} + \frac{218385042750}{1417708231} a^{7} + \frac{147002469075}{1417708231} a^{6} + \frac{119405553225}{1417708231} a^{5} + \frac{49544069400}{1417708231} a^{4} + \frac{58447871898}{1417708231} a^{3} + \frac{3911373900}{1417708231} a^{2} - \frac{760544925}{1417708231} a + \frac{434597100}{1417708231} \) (order $10$)
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:  \( 2526424.45141 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_5$ (as 20T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.160801.1, 10.10.80803005003125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
401Data not computed