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Properties

Label 20.0.19000149661...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 53^{8}$
Root discriminant $16.37$
Ramified primes $5, 53$
Class number $1$
Class group Trivial
Galois group $C_4\times A_5$ (as 20T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 6, 4, 5, 6, -1, 11, 35, 9, -24, -6, 5, 11, 4, -9, 0, -1, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - x^17 - 9*x^15 + 4*x^14 + 11*x^13 + 5*x^12 - 6*x^11 - 24*x^10 + 9*x^9 + 35*x^8 + 11*x^7 - x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 6*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + x^18 - x^17 - 9*x^15 + 4*x^14 + 11*x^13 + 5*x^12 - 6*x^11 - 24*x^10 + 9*x^9 + 35*x^8 + 11*x^7 - x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 6*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + x^{18} - x^{17} - 9 x^{15} + 4 x^{14} + 11 x^{13} + 5 x^{12} - 6 x^{11} - 24 x^{10} + 9 x^{9} + 35 x^{8} + 11 x^{7} - x^{6} + 6 x^{5} + 5 x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1 \)

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:  \(1900014966167022705078125=5^{15}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.37$
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:  $5, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3001} a^{18} + \frac{886}{3001} a^{17} + \frac{418}{3001} a^{16} - \frac{452}{3001} a^{15} + \frac{1245}{3001} a^{14} - \frac{186}{3001} a^{13} - \frac{989}{3001} a^{12} + \frac{868}{3001} a^{11} - \frac{306}{3001} a^{10} + \frac{228}{3001} a^{9} + \frac{344}{3001} a^{8} + \frac{691}{3001} a^{7} - \frac{1176}{3001} a^{6} - \frac{210}{3001} a^{5} - \frac{1169}{3001} a^{4} - \frac{866}{3001} a^{3} - \frac{1264}{3001} a^{2} + \frac{1238}{3001} a + \frac{674}{3001}$, $\frac{1}{23980991} a^{19} - \frac{103}{23980991} a^{18} + \frac{8265210}{23980991} a^{17} - \frac{4252133}{23980991} a^{16} - \frac{737122}{23980991} a^{15} - \frac{10828689}{23980991} a^{14} - \frac{9339208}{23980991} a^{13} - \frac{7231747}{23980991} a^{12} + \frac{7844142}{23980991} a^{11} + \frac{10017099}{23980991} a^{10} + \frac{1503428}{23980991} a^{9} + \frac{2091285}{23980991} a^{8} - \frac{11602213}{23980991} a^{7} + \frac{4881093}{23980991} a^{6} + \frac{10499951}{23980991} a^{5} + \frac{1293321}{23980991} a^{4} + \frac{2577784}{23980991} a^{3} - \frac{1380543}{23980991} a^{2} - \frac{5284061}{23980991} a - \frac{9168419}{23980991}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{113230098}{23980991} a^{19} - \frac{217294677}{23980991} a^{18} + \frac{305441318}{23980991} a^{17} - \frac{374399254}{23980991} a^{16} + \frac{315786949}{23980991} a^{15} - \frac{1273394608}{23980991} a^{14} + \frac{1589089382}{23980991} a^{13} - \frac{122550440}{23980991} a^{12} + \frac{521208488}{23980991} a^{11} - \frac{1100751822}{23980991} a^{10} - \frac{1723166886}{23980991} a^{9} + \frac{2701019032}{23980991} a^{8} + \frac{1574391426}{23980991} a^{7} - \frac{488977743}{23980991} a^{6} + \frac{317196304}{23980991} a^{5} + \frac{507368042}{23980991} a^{4} + \frac{99593619}{23980991} a^{3} + \frac{371214038}{23980991} a^{2} + \frac{336586004}{23980991} a + \frac{121406613}{23980991} \) (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:  \( 31104.0765784 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_5$ (as 20T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 20 conjugacy class representatives for $C_4\times A_5$
Character table for $C_4\times A_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.70225.1, 10.2.24657753125.1

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

Sibling fields

Degree 24 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $20$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$53$53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.12.8.1$x^{12} - 159 x^{9} + 8427 x^{6} - 148877 x^{3} + 46017285192$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$