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Properties

Label 20.0.32056524103...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 13^{10}\cdot 47^{8}$
Root discriminant $37.61$
Ramified primes $5, 13, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_5\times A_4$ (as 20T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, -576, 2400, -8, -1341, 2192, -1174, 246, 359, -321, 47, 137, -126, 39, 4, -6, 2, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 6*x^16 + 4*x^15 + 39*x^14 - 126*x^13 + 137*x^12 + 47*x^11 - 321*x^10 + 359*x^9 + 246*x^8 - 1174*x^7 + 2192*x^6 - 1341*x^5 - 8*x^4 + 2400*x^3 - 576*x^2 + 1024)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 6*x^16 + 4*x^15 + 39*x^14 - 126*x^13 + 137*x^12 + 47*x^11 - 321*x^10 + 359*x^9 + 246*x^8 - 1174*x^7 + 2192*x^6 - 1341*x^5 - 8*x^4 + 2400*x^3 - 576*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} - 6 x^{16} + 4 x^{15} + 39 x^{14} - 126 x^{13} + 137 x^{12} + 47 x^{11} - 321 x^{10} + 359 x^{9} + 246 x^{8} - 1174 x^{7} + 2192 x^{6} - 1341 x^{5} - 8 x^{4} + 2400 x^{3} - 576 x^{2} + 1024 \)

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:  \(32056524103267394907090712890625=5^{10}\cdot 13^{10}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.61$
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:  $5, 13, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{16} + \frac{1}{8} a^{15} + \frac{1}{8} a^{14} - \frac{3}{8} a^{13} + \frac{1}{4} a^{12} + \frac{7}{16} a^{11} + \frac{1}{8} a^{10} - \frac{7}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{7}{16} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{18} - \frac{3}{160} a^{17} - \frac{19}{160} a^{16} - \frac{51}{160} a^{15} - \frac{23}{160} a^{14} - \frac{1}{80} a^{13} + \frac{119}{320} a^{12} - \frac{13}{160} a^{11} - \frac{3}{64} a^{10} - \frac{149}{320} a^{9} - \frac{9}{64} a^{8} + \frac{31}{64} a^{7} - \frac{27}{160} a^{6} + \frac{1}{160} a^{5} - \frac{3}{8} a^{4} - \frac{29}{320} a^{3} + \frac{27}{80} a^{2} - \frac{1}{20} a - \frac{1}{5}$, $\frac{1}{1761209314916154230754799360} a^{19} + \frac{467392082733546141337597}{880604657458077115377399680} a^{18} + \frac{7885248669069153475731021}{880604657458077115377399680} a^{17} - \frac{34673652697405383666006291}{880604657458077115377399680} a^{16} - \frac{134822691770212764894947063}{880604657458077115377399680} a^{15} - \frac{51817415686994717092127601}{440302328729038557688699840} a^{14} - \frac{445960966306124946077127241}{1761209314916154230754799360} a^{13} - \frac{398588229129950434768628333}{880604657458077115377399680} a^{12} + \frac{81820275257976713947246973}{352241862983230846150959872} a^{11} - \frac{602361331121304849901092309}{1761209314916154230754799360} a^{10} + \frac{84777910783161742359361335}{352241862983230846150959872} a^{9} - \frac{65504057363297806357104801}{352241862983230846150959872} a^{8} - \frac{282916561075099753662623387}{880604657458077115377399680} a^{7} + \frac{47526454560227594492450881}{880604657458077115377399680} a^{6} + \frac{16841305378538538876759573}{44030232872903855768869984} a^{5} - \frac{466135210437240011695634909}{1761209314916154230754799360} a^{4} - \frac{37866403025932346975270533}{440302328729038557688699840} a^{3} + \frac{53985282309373105372296819}{110075582182259639422174960} a^{2} + \frac{7797178657940961360950279}{27518895545564909855543740} a - \frac{475159828770040448975959}{1375944777278245492777187}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 14363449.2462 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\times A_4$ (as 20T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 16 conjugacy class representatives for $D_5\times A_4$
Character table for $D_5\times A_4$

Intermediate fields

4.0.4225.1, 5.1.2209.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$