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Properties

Label 20.0.39220259306...0481.1
Degree $20$
Signature $[0, 10]$
Discriminant $191^{2}\cdot 401^{10}$
Root discriminant $33.86$
Ramified primes $191, 401$
Class number $17$ (GRH)
Class group $[17]$ (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, -682, 1146, -394, -708, -492, 4107, -6630, 7229, -6402, 4808, -3452, 2560, -1744, 962, -460, 236, -120, 45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 236*x^16 - 460*x^15 + 962*x^14 - 1744*x^13 + 2560*x^12 - 3452*x^11 + 4808*x^10 - 6402*x^9 + 7229*x^8 - 6630*x^7 + 4107*x^6 - 492*x^5 - 708*x^4 - 394*x^3 + 1146*x^2 - 682*x + 961)
 
gp: K = bnfinit(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 236*x^16 - 460*x^15 + 962*x^14 - 1744*x^13 + 2560*x^12 - 3452*x^11 + 4808*x^10 - 6402*x^9 + 7229*x^8 - 6630*x^7 + 4107*x^6 - 492*x^5 - 708*x^4 - 394*x^3 + 1146*x^2 - 682*x + 961, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 45 x^{18} - 120 x^{17} + 236 x^{16} - 460 x^{15} + 962 x^{14} - 1744 x^{13} + 2560 x^{12} - 3452 x^{11} + 4808 x^{10} - 6402 x^{9} + 7229 x^{8} - 6630 x^{7} + 4107 x^{6} - 492 x^{5} - 708 x^{4} - 394 x^{3} + 1146 x^{2} - 682 x + 961 \)

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:  \(3922025930637479853172889160481=191^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.86$
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:  $191, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{15} - \frac{2}{9} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{6} a^{5} - \frac{7}{18} a^{4} - \frac{7}{18} a^{3} + \frac{7}{18} a^{2} - \frac{1}{3} a + \frac{1}{18}$, $\frac{1}{18} a^{17} - \frac{1}{18} a^{15} - \frac{2}{9} a^{14} + \frac{1}{18} a^{13} - \frac{1}{6} a^{12} + \frac{2}{9} a^{11} + \frac{1}{18} a^{10} - \frac{5}{18} a^{9} + \frac{2}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{18} a^{5} - \frac{1}{2} a^{4} + \frac{5}{18} a^{3} - \frac{2}{9} a^{2} + \frac{7}{18} a + \frac{4}{9}$, $\frac{1}{4209109465266} a^{18} - \frac{1}{467678829474} a^{17} + \frac{111950440249}{4209109465266} a^{16} - \frac{194085277577}{4209109465266} a^{15} - \frac{387649433254}{2104554732633} a^{14} + \frac{27303184901}{2104554732633} a^{13} - \frac{883892093825}{4209109465266} a^{12} + \frac{161229107897}{2104554732633} a^{11} - \frac{26979639611}{1403036488422} a^{10} + \frac{358153983301}{2104554732633} a^{9} + \frac{1431996923965}{4209109465266} a^{8} - \frac{230865940655}{701518244211} a^{7} + \frac{406782898709}{1403036488422} a^{6} + \frac{445488101189}{1403036488422} a^{5} + \frac{22384616392}{701518244211} a^{4} + \frac{602167142777}{1403036488422} a^{3} - \frac{299073674798}{701518244211} a^{2} - \frac{1319570949109}{4209109465266} a - \frac{54727747705}{135777724686}$, $\frac{1}{41371336934099514} a^{19} + \frac{545}{4596815214899946} a^{18} + \frac{1100560236148345}{41371336934099514} a^{17} + \frac{389048860766953}{41371336934099514} a^{16} - \frac{840641626941241}{20685668467049757} a^{15} + \frac{2146853527322744}{20685668467049757} a^{14} + \frac{3623515089418627}{41371336934099514} a^{13} + \frac{3132617799059243}{20685668467049757} a^{12} + \frac{613900696567861}{13790445644699838} a^{11} - \frac{5934568719565657}{41371336934099514} a^{10} + \frac{7856527455474581}{20685668467049757} a^{9} - \frac{673041939260882}{6895222822349919} a^{8} - \frac{5911449977574775}{13790445644699838} a^{7} - \frac{5467583539024669}{13790445644699838} a^{6} - \frac{29964095530126}{130098543817923} a^{5} + \frac{586668949576735}{6895222822349919} a^{4} - \frac{5189995000140499}{13790445644699838} a^{3} + \frac{801413402416313}{41371336934099514} a^{2} - \frac{10428133057736827}{41371336934099514} a - \frac{554586865571}{16476040196774}$

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

Class group and class number

$C_{17}$, which has order $17$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.0.1980410545982191.1, 10.0.4938679665791.1, 10.10.10368641602001.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$191$191.4.0.1$x^{4} - x + 28$$1$$4$$0$$C_4$$[\ ]^{4}$
191.4.0.1$x^{4} - x + 28$$1$$4$$0$$C_4$$[\ ]^{4}$
191.4.0.1$x^{4} - x + 28$$1$$4$$0$$C_4$$[\ ]^{4}$
191.4.0.1$x^{4} - x + 28$$1$$4$$0$$C_4$$[\ ]^{4}$
191.4.2.1$x^{4} + 7067 x^{2} + 13169641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed