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Properties

Label 20.20.1005420215...4353.1
Degree $20$
Signature $[20, 0]$
Discriminant $17^{15}\cdot 37^{8}$
Root discriminant $35.49$
Ramified primes $17, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -17, 22, 657, -1888, -2752, 14111, -8176, -20370, 25627, 3882, -18033, 4019, 5370, -2109, -758, 398, 48, -33, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 33*x^18 + 48*x^17 + 398*x^16 - 758*x^15 - 2109*x^14 + 5370*x^13 + 4019*x^12 - 18033*x^11 + 3882*x^10 + 25627*x^9 - 20370*x^8 - 8176*x^7 + 14111*x^6 - 2752*x^5 - 1888*x^4 + 657*x^3 + 22*x^2 - 17*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 33*x^18 + 48*x^17 + 398*x^16 - 758*x^15 - 2109*x^14 + 5370*x^13 + 4019*x^12 - 18033*x^11 + 3882*x^10 + 25627*x^9 - 20370*x^8 - 8176*x^7 + 14111*x^6 - 2752*x^5 - 1888*x^4 + 657*x^3 + 22*x^2 - 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 33 x^{18} + 48 x^{17} + 398 x^{16} - 758 x^{15} - 2109 x^{14} + 5370 x^{13} + 4019 x^{12} - 18033 x^{11} + 3882 x^{10} + 25627 x^{9} - 20370 x^{8} - 8176 x^{7} + 14111 x^{6} - 2752 x^{5} - 1888 x^{4} + 657 x^{3} + 22 x^{2} - 17 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10054202156858080231167941574353=17^{15}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.49$
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:  $17, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{51} a^{16} + \frac{11}{51} a^{15} - \frac{2}{17} a^{14} - \frac{1}{3} a^{13} - \frac{7}{51} a^{12} - \frac{6}{17} a^{11} + \frac{1}{51} a^{10} - \frac{4}{17} a^{9} + \frac{13}{51} a^{8} - \frac{23}{51} a^{7} - \frac{20}{51} a^{6} + \frac{20}{51} a^{5} + \frac{23}{51} a^{4} + \frac{22}{51} a^{3} - \frac{19}{51} a^{2} - \frac{2}{17} a - \frac{13}{51}$, $\frac{1}{51} a^{17} - \frac{25}{51} a^{15} - \frac{2}{51} a^{14} - \frac{8}{17} a^{13} + \frac{8}{51} a^{12} - \frac{5}{51} a^{11} - \frac{23}{51} a^{10} - \frac{8}{51} a^{9} - \frac{13}{51} a^{8} - \frac{22}{51} a^{7} - \frac{5}{17} a^{6} + \frac{7}{51} a^{5} + \frac{8}{17} a^{4} - \frac{2}{17} a^{3} - \frac{1}{51} a^{2} + \frac{2}{51} a - \frac{10}{51}$, $\frac{1}{663} a^{18} + \frac{1}{663} a^{17} + \frac{1}{221} a^{16} + \frac{179}{663} a^{15} - \frac{11}{51} a^{14} - \frac{96}{221} a^{13} + \frac{266}{663} a^{12} - \frac{328}{663} a^{11} + \frac{84}{221} a^{10} + \frac{3}{13} a^{9} - \frac{283}{663} a^{8} - \frac{74}{221} a^{7} - \frac{211}{663} a^{6} + \frac{6}{17} a^{5} + \frac{203}{663} a^{4} + \frac{50}{221} a^{3} - \frac{92}{221} a^{2} - \frac{125}{663} a + \frac{2}{39}$, $\frac{1}{7714921122129} a^{19} + \frac{3357623714}{7714921122129} a^{18} + \frac{17772988433}{7714921122129} a^{17} - \frac{56205319037}{7714921122129} a^{16} - \frac{417007305366}{2571640374043} a^{15} + \frac{2012109945194}{7714921122129} a^{14} + \frac{50478622565}{453818889537} a^{13} - \frac{423548909996}{7714921122129} a^{12} + \frac{390835580620}{2571640374043} a^{11} - \frac{1135539918554}{2571640374043} a^{10} - \frac{739940005676}{2571640374043} a^{9} - \frac{1209922828888}{2571640374043} a^{8} + \frac{1065567768493}{2571640374043} a^{7} + \frac{1082749102663}{7714921122129} a^{6} - \frac{2227906084721}{7714921122129} a^{5} - \frac{1143187996913}{2571640374043} a^{4} + \frac{3003872460392}{7714921122129} a^{3} + \frac{688548309643}{7714921122129} a^{2} + \frac{3591342376549}{7714921122129} a + \frac{3433259476975}{7714921122129}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 1110509306.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 5.5.6725897.1, 10.10.769040737728353.1

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 10 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$