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Properties

Label 20.20.7919632171...1209.1
Degree $20$
Signature $[20, 0]$
Discriminant $19^{8}\cdot 293^{10}$
Root discriminant $55.58$
Ramified primes $19, 293$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times A_5$ (as 20T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3877, 2654, -142212, 150931, 484524, -352394, -658303, 320340, 477144, -149409, -204919, 38628, 54107, -5526, -8787, 407, 851, -12, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 45*x^18 - 12*x^17 + 851*x^16 + 407*x^15 - 8787*x^14 - 5526*x^13 + 54107*x^12 + 38628*x^11 - 204919*x^10 - 149409*x^9 + 477144*x^8 + 320340*x^7 - 658303*x^6 - 352394*x^5 + 484524*x^4 + 150931*x^3 - 142212*x^2 + 2654*x + 3877)
 
gp: K = bnfinit(x^20 - 45*x^18 - 12*x^17 + 851*x^16 + 407*x^15 - 8787*x^14 - 5526*x^13 + 54107*x^12 + 38628*x^11 - 204919*x^10 - 149409*x^9 + 477144*x^8 + 320340*x^7 - 658303*x^6 - 352394*x^5 + 484524*x^4 + 150931*x^3 - 142212*x^2 + 2654*x + 3877, 1)
 

Normalized defining polynomial

\( x^{20} - 45 x^{18} - 12 x^{17} + 851 x^{16} + 407 x^{15} - 8787 x^{14} - 5526 x^{13} + 54107 x^{12} + 38628 x^{11} - 204919 x^{10} - 149409 x^{9} + 477144 x^{8} + 320340 x^{7} - 658303 x^{6} - 352394 x^{5} + 484524 x^{4} + 150931 x^{3} - 142212 x^{2} + 2654 x + 3877 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(79196321710980185081864322433061209=19^{8}\cdot 293^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.58$
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:  $19, 293$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{85555676540574218170512823} a^{19} - \frac{16291343449855732584498519}{85555676540574218170512823} a^{18} - \frac{21374564625197955784787330}{85555676540574218170512823} a^{17} + \frac{34666476078749723274300480}{85555676540574218170512823} a^{16} - \frac{32233324765721597943360174}{85555676540574218170512823} a^{15} + \frac{15628636812277879644217707}{85555676540574218170512823} a^{14} + \frac{41562273078914119627206968}{85555676540574218170512823} a^{13} + \frac{41140168496024025579506428}{85555676540574218170512823} a^{12} - \frac{9359830429466685822056210}{85555676540574218170512823} a^{11} + \frac{29144257691678376666113813}{85555676540574218170512823} a^{10} - \frac{40434879575244787203006332}{85555676540574218170512823} a^{9} + \frac{42066874359269221142057582}{85555676540574218170512823} a^{8} - \frac{38359390512533165424941996}{85555676540574218170512823} a^{7} - \frac{12964157448658785634559891}{85555676540574218170512823} a^{6} + \frac{1426771039759797584354008}{85555676540574218170512823} a^{5} + \frac{20436348709095908799786368}{85555676540574218170512823} a^{4} - \frac{12989044573913558870712539}{85555676540574218170512823} a^{3} + \frac{1850988620338692464601635}{85555676540574218170512823} a^{2} - \frac{21326625385965807100729175}{85555676540574218170512823} a - \frac{15794816270122884245771278}{85555676540574218170512823}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 96962157521.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{293}) \), 10.10.960472390437121.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
19.4.2.1$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}$
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}$
293Data not computed