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Properties

Label 20.0.55180948378...3632.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 11^{10}$
Root discriminant $38.64$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, -5184, -7344, 45792, -5868, -51264, 135888, -158688, 173896, -71232, 23124, -2824, -43, -184, 1232, -880, 462, -168, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 44*x^18 - 168*x^17 + 462*x^16 - 880*x^15 + 1232*x^14 - 184*x^13 - 43*x^12 - 2824*x^11 + 23124*x^10 - 71232*x^9 + 173896*x^8 - 158688*x^7 + 135888*x^6 - 51264*x^5 - 5868*x^4 + 45792*x^3 - 7344*x^2 - 5184*x + 1296)
 
gp: K = bnfinit(x^20 - 8*x^19 + 44*x^18 - 168*x^17 + 462*x^16 - 880*x^15 + 1232*x^14 - 184*x^13 - 43*x^12 - 2824*x^11 + 23124*x^10 - 71232*x^9 + 173896*x^8 - 158688*x^7 + 135888*x^6 - 51264*x^5 - 5868*x^4 + 45792*x^3 - 7344*x^2 - 5184*x + 1296, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 44 x^{18} - 168 x^{17} + 462 x^{16} - 880 x^{15} + 1232 x^{14} - 184 x^{13} - 43 x^{12} - 2824 x^{11} + 23124 x^{10} - 71232 x^{9} + 173896 x^{8} - 158688 x^{7} + 135888 x^{6} - 51264 x^{5} - 5868 x^{4} + 45792 x^{3} - 7344 x^{2} - 5184 x + 1296 \)

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:  \(55180948378603639203880482373632=2^{55}\cdot 3^{10}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.64$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{24} a^{12} + \frac{1}{24} a^{10} - \frac{1}{6} a^{9} + \frac{1}{8} a^{8} + \frac{5}{24} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} + \frac{5}{12} a^{2} - \frac{1}{2}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} + \frac{1}{48} a^{11} - \frac{5}{48} a^{10} + \frac{7}{48} a^{9} - \frac{1}{16} a^{8} + \frac{5}{48} a^{7} + \frac{7}{48} a^{6} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{11}{24} a^{3} - \frac{5}{24} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{48} a^{14} + \frac{1}{24} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{5}{24} a^{8} - \frac{1}{8} a^{7} - \frac{11}{48} a^{6} - \frac{1}{3} a^{5} - \frac{1}{24} a^{4} - \frac{5}{12} a^{3} + \frac{7}{24} a^{2} + \frac{1}{4}$, $\frac{1}{48} a^{15} + \frac{1}{24} a^{11} - \frac{1}{12} a^{10} + \frac{5}{24} a^{9} - \frac{5}{48} a^{7} + \frac{1}{12} a^{6} + \frac{5}{24} a^{5} - \frac{1}{6} a^{4} + \frac{5}{24} a^{3} - \frac{1}{6} a^{2} - \frac{1}{4} a$, $\frac{1}{3168} a^{16} - \frac{7}{1584} a^{15} - \frac{1}{144} a^{14} + \frac{1}{132} a^{13} + \frac{1}{132} a^{12} + \frac{1}{99} a^{11} + \frac{7}{99} a^{10} - \frac{67}{792} a^{9} - \frac{29}{288} a^{8} - \frac{71}{1584} a^{7} - \frac{3}{176} a^{6} + \frac{35}{88} a^{5} + \frac{299}{1584} a^{4} + \frac{23}{88} a^{3} + \frac{5}{264} a^{2} - \frac{19}{44} a - \frac{1}{11}$, $\frac{1}{9504} a^{17} + \frac{1}{9504} a^{16} + \frac{1}{297} a^{15} + \frac{5}{528} a^{14} - \frac{1}{792} a^{13} + \frac{4}{297} a^{12} + \frac{1}{216} a^{11} - \frac{151}{2376} a^{10} + \frac{1337}{9504} a^{9} - \frac{1759}{9504} a^{8} + \frac{137}{792} a^{7} - \frac{35}{1584} a^{6} - \frac{811}{4752} a^{5} + \frac{31}{528} a^{4} + \frac{179}{396} a^{3} + \frac{53}{264} a^{2} - \frac{47}{132} a + \frac{1}{22}$, $\frac{1}{133056} a^{18} - \frac{1}{22176} a^{17} - \frac{5}{33264} a^{16} + \frac{31}{8316} a^{15} + \frac{31}{5544} a^{14} + \frac{179}{33264} a^{13} + \frac{169}{11088} a^{12} - \frac{295}{11088} a^{11} - \frac{53}{44352} a^{10} - \frac{1031}{22176} a^{9} - \frac{1151}{8316} a^{8} - \frac{251}{1232} a^{7} - \frac{3613}{66528} a^{6} - \frac{15751}{33264} a^{5} - \frac{907}{5544} a^{4} + \frac{1003}{5544} a^{3} + \frac{27}{154} a^{2} + \frac{149}{462} a + \frac{1}{308}$, $\frac{1}{7001891374746957877964456284089024} a^{19} + \frac{2332648858278540250793176121}{3500945687373478938982228142044512} a^{18} - \frac{12263568275931288066996728629}{1750472843686739469491114071022256} a^{17} - \frac{27260926530896619676867158433}{194496982631859941054568230113584} a^{16} + \frac{270267528390988706941338642109}{36468184243473738947731543146297} a^{15} + \frac{12591507895861484197020715497743}{1750472843686739469491114071022256} a^{14} - \frac{8522936122492522546830861799487}{875236421843369734745557035511128} a^{13} - \frac{14801913955159005748504423495115}{875236421843369734745557035511128} a^{12} - \frac{94677790458819969906404337027883}{7001891374746957877964456284089024} a^{11} - \frac{185075450465552410837996193014607}{3500945687373478938982228142044512} a^{10} + \frac{1387479899018606332586985130907}{27785283233122848722081175730512} a^{9} + \frac{114456182663143206900727567198957}{583490947895579823163704690340752} a^{8} - \frac{72741313277967418057626766220857}{500135098196211276997461163149216} a^{7} + \frac{7820823140971067945547412009727}{291745473947789911581852345170376} a^{6} + \frac{48403803689642110375392294849343}{291745473947789911581852345170376} a^{5} - \frac{6330931013547680727384423441893}{16208081885988328421214019176132} a^{4} + \frac{9691787041994934269142405016829}{32416163771976656842428038352264} a^{3} + \frac{32863285133260314172567610673}{140329713298600246071117049144} a^{2} - \frac{73290995794685817000402289987}{736730994817651291873364508006} a - \frac{2274902389151986278311172336727}{5402693961996109473738006392044}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 298817472.483 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.2230272.1, 5.1.2230272.1 x5, 10.2.39792905551872.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.2230272.1
Degree 10 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$