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Properties

Label 20.0.33138892049...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 19^{4}\cdot 1699^{4}$
Root discriminant $26.67$
Ramified primes $5, 19, 1699$
Class number $9$
Class group $[3, 3]$
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, -4, 16, -64, 302, 178, 90, -143, 525, 10, 433, 70, 292, -33, 140, -57, 60, -16, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 11*x^18 - 16*x^17 + 60*x^16 - 57*x^15 + 140*x^14 - 33*x^13 + 292*x^12 + 70*x^11 + 433*x^10 + 10*x^9 + 525*x^8 - 143*x^7 + 90*x^6 + 178*x^5 + 302*x^4 - 64*x^3 + 16*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 11*x^18 - 16*x^17 + 60*x^16 - 57*x^15 + 140*x^14 - 33*x^13 + 292*x^12 + 70*x^11 + 433*x^10 + 10*x^9 + 525*x^8 - 143*x^7 + 90*x^6 + 178*x^5 + 302*x^4 - 64*x^3 + 16*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 11 x^{18} - 16 x^{17} + 60 x^{16} - 57 x^{15} + 140 x^{14} - 33 x^{13} + 292 x^{12} + 70 x^{11} + 433 x^{10} + 10 x^{9} + 525 x^{8} - 143 x^{7} + 90 x^{6} + 178 x^{5} + 302 x^{4} - 64 x^{3} + 16 x^{2} - 4 x + 1 \)

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:  \(33138892049807358428955078125=5^{15}\cdot 19^{4}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.67$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{19} a^{17} - \frac{2}{19} a^{16} - \frac{1}{19} a^{15} + \frac{5}{19} a^{14} + \frac{9}{19} a^{13} + \frac{5}{19} a^{12} + \frac{7}{19} a^{11} + \frac{8}{19} a^{10} - \frac{6}{19} a^{9} + \frac{3}{19} a^{8} - \frac{6}{19} a^{7} + \frac{3}{19} a^{6} - \frac{7}{19} a^{5} + \frac{3}{19} a^{4} + \frac{9}{19} a^{3} - \frac{7}{19} a^{2} + \frac{7}{19} a - \frac{7}{19}$, $\frac{1}{19} a^{18} - \frac{5}{19} a^{16} + \frac{3}{19} a^{15} + \frac{4}{19} a^{13} - \frac{2}{19} a^{12} + \frac{3}{19} a^{11} - \frac{9}{19} a^{10} - \frac{9}{19} a^{9} - \frac{9}{19} a^{7} - \frac{1}{19} a^{6} + \frac{8}{19} a^{5} - \frac{4}{19} a^{4} - \frac{8}{19} a^{3} - \frac{7}{19} a^{2} + \frac{7}{19} a + \frac{5}{19}$, $\frac{1}{29304915596449678980488851} a^{19} + \frac{102531819636152367152579}{29304915596449678980488851} a^{18} - \frac{526324050053995386304485}{29304915596449678980488851} a^{17} - \frac{14513141683633034815541128}{29304915596449678980488851} a^{16} + \frac{2309401270547959410085996}{29304915596449678980488851} a^{15} + \frac{9463393095705097648831840}{29304915596449678980488851} a^{14} - \frac{12789804581555054243311915}{29304915596449678980488851} a^{13} + \frac{652322598563248712931780}{29304915596449678980488851} a^{12} - \frac{3377583672033985270108764}{29304915596449678980488851} a^{11} - \frac{3068306591570220676264613}{29304915596449678980488851} a^{10} - \frac{518605252356398821530278}{1542363978760509420025729} a^{9} - \frac{10357356287356923643939748}{29304915596449678980488851} a^{8} + \frac{9193516786927077177047192}{29304915596449678980488851} a^{7} + \frac{5543824684969277057880758}{29304915596449678980488851} a^{6} - \frac{14588690905246231155787092}{29304915596449678980488851} a^{5} + \frac{13080436558478857817408046}{29304915596449678980488851} a^{4} - \frac{789845524303905783594573}{29304915596449678980488851} a^{3} - \frac{8231722222824461595821421}{29304915596449678980488851} a^{2} + \frac{9917602022142655844382780}{29304915596449678980488851} a - \frac{6481788387558187266972681}{29304915596449678980488851}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:  \( \frac{443804534804568912492857}{29304915596449678980488851} a^{19} - \frac{1760942791456694095685856}{29304915596449678980488851} a^{18} + \frac{5841957704907681420998578}{29304915596449678980488851} a^{17} - \frac{1116264293210173903701896}{1542363978760509420025729} a^{16} + \frac{43770421141504709440258998}{29304915596449678980488851} a^{15} - \frac{98397320397255039238058967}{29304915596449678980488851} a^{14} + \frac{115651430484503575032639745}{29304915596449678980488851} a^{13} - \frac{177863522979880153594216159}{29304915596449678980488851} a^{12} + \frac{6490591451829598149365098}{1542363978760509420025729} a^{11} - \frac{17958504611600802927763616}{1542363978760509420025729} a^{10} - \frac{591439235330514181361618}{29304915596449678980488851} a^{9} - \frac{589040203648510350604936895}{29304915596449678980488851} a^{8} + \frac{74191274229571598296902484}{29304915596449678980488851} a^{7} - \frac{755555257197447011558265065}{29304915596449678980488851} a^{6} + \frac{48571545934154744545903338}{29304915596449678980488851} a^{5} + \frac{10115712902197362521075687}{29304915596449678980488851} a^{4} - \frac{127391129250472761559858317}{29304915596449678980488851} a^{3} - \frac{525507536744540711476161671}{29304915596449678980488851} a^{2} - \frac{404246028096363746706362}{1542363978760509420025729} a + \frac{159350305143406578514537}{29304915596449678980488851} \) (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:  \( 351148.02443 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.10.3256446753125.1

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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1699Data not computed