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Properties

Label 20.0.38849915667...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 3359^{4}$
Root discriminant $16.96$
Ramified primes $5, 3359$
Class number $1$
Class group Trivial
Galois group $D_5^2:C_4$ (as 20T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 19, 17, 46, 27, 42, 14, 11, -20, 9, -39, 11, -15, 10, -2, 11, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 + 11*x^16 - 2*x^15 + 10*x^14 - 15*x^13 + 11*x^12 - 39*x^11 + 9*x^10 - 20*x^9 + 11*x^8 + 14*x^7 + 42*x^6 + 27*x^5 + 46*x^4 + 17*x^3 + 19*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 + 5*x^18 + 11*x^16 - 2*x^15 + 10*x^14 - 15*x^13 + 11*x^12 - 39*x^11 + 9*x^10 - 20*x^9 + 11*x^8 + 14*x^7 + 42*x^6 + 27*x^5 + 46*x^4 + 17*x^3 + 19*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} + 11 x^{16} - 2 x^{15} + 10 x^{14} - 15 x^{13} + 11 x^{12} - 39 x^{11} + 9 x^{10} - 20 x^{9} + 11 x^{8} + 14 x^{7} + 42 x^{6} + 27 x^{5} + 46 x^{4} + 17 x^{3} + 19 x^{2} + 3 x + 1 \)

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:  \(3884991566777374267578125=5^{15}\cdot 3359^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.96$
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:  $5, 3359$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11206186407419} a^{19} + \frac{1382357461060}{11206186407419} a^{18} - \frac{3661493474063}{11206186407419} a^{17} - \frac{2498688731524}{11206186407419} a^{16} + \frac{4656349804309}{11206186407419} a^{15} + \frac{1371770106011}{11206186407419} a^{14} + \frac{4815965575435}{11206186407419} a^{13} - \frac{689093615574}{11206186407419} a^{12} + \frac{5055110032032}{11206186407419} a^{11} + \frac{5117175536039}{11206186407419} a^{10} - \frac{5346973405207}{11206186407419} a^{9} + \frac{1786190023765}{11206186407419} a^{8} - \frac{4319186207330}{11206186407419} a^{7} - \frac{3092354891830}{11206186407419} a^{6} + \frac{1818846296855}{11206186407419} a^{5} + \frac{3961179661248}{11206186407419} a^{4} + \frac{3853714335089}{11206186407419} a^{3} - \frac{4212794731954}{11206186407419} a^{2} - \frac{490280981011}{11206186407419} a - \frac{3576886184860}{11206186407419}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{110528629712}{11206186407419} a^{19} + \frac{93576025357}{11206186407419} a^{18} + \frac{1837453616367}{11206186407419} a^{17} + \frac{701703112967}{11206186407419} a^{16} + \frac{6642766518714}{11206186407419} a^{15} + \frac{1755114373184}{11206186407419} a^{14} + \frac{10727628967951}{11206186407419} a^{13} - \frac{1178645381313}{11206186407419} a^{12} + \frac{3796355546793}{11206186407419} a^{11} - \frac{18598763311565}{11206186407419} a^{10} + \frac{2148468728584}{11206186407419} a^{9} - \frac{34518886138706}{11206186407419} a^{8} - \frac{6980655423298}{11206186407419} a^{7} + \frac{7551198518324}{11206186407419} a^{6} + \frac{11876955850778}{11206186407419} a^{5} + \frac{36349521736472}{11206186407419} a^{4} + \frac{48666132807425}{11206186407419} a^{3} + \frac{34621167432625}{11206186407419} a^{2} + \frac{33616954797511}{11206186407419} a + \frac{12928134974155}{11206186407419} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 34252.1729165 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T93):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.6.35259003125.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
3359Data not computed