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Properties

Label 20.0.17440473950...2736.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{32}\cdot 67^{8}$
Root discriminant $16.30$
Ramified primes $2, 67$
Class number $1$
Class group Trivial
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![4, -16, 32, -40, 48, -76, 120, -136, 109, -66, 34, -4, -22, 30, -16, 2, 5, -8, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*x + 4)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 8 x^{17} + 5 x^{16} + 2 x^{15} - 16 x^{14} + 30 x^{13} - 22 x^{12} - 4 x^{11} + 34 x^{10} - 66 x^{9} + 109 x^{8} - 136 x^{7} + 120 x^{6} - 76 x^{5} + 48 x^{4} - 40 x^{3} + 32 x^{2} - 16 x + 4 \)

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:  \(1744047395068446282612736=2^{32}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.30$
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:  $2, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{159851548} a^{19} - \frac{4703375}{39962887} a^{18} + \frac{6748419}{79925774} a^{17} - \frac{2354475}{159851548} a^{16} - \frac{76917031}{159851548} a^{15} - \frac{5664755}{39962887} a^{14} - \frac{1514615}{79925774} a^{13} + \frac{12706489}{159851548} a^{12} - \frac{11427671}{39962887} a^{11} + \frac{10227295}{79925774} a^{10} + \frac{8360643}{39962887} a^{9} + \frac{1043761}{79925774} a^{8} + \frac{20328221}{159851548} a^{7} + \frac{2022511}{79925774} a^{6} + \frac{198553}{39962887} a^{5} + \frac{19349637}{159851548} a^{4} - \frac{18012177}{39962887} a^{3} - \frac{28244355}{79925774} a^{2} - \frac{9619111}{39962887} a - \frac{20147829}{79925774}$

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:  \( \frac{26758413}{159851548} a^{19} - \frac{60872855}{79925774} a^{18} + \frac{122020429}{79925774} a^{17} - \frac{121676233}{79925774} a^{16} + \frac{115506207}{159851548} a^{15} + \frac{15963377}{39962887} a^{14} - \frac{251929171}{79925774} a^{13} + \frac{239526437}{39962887} a^{12} - \frac{159792325}{39962887} a^{11} - \frac{53142461}{39962887} a^{10} + \frac{563142481}{79925774} a^{9} - \frac{993516103}{79925774} a^{8} + \frac{3289992433}{159851548} a^{7} - \frac{1993748639}{79925774} a^{6} + \frac{1672955167}{79925774} a^{5} - \frac{889058877}{79925774} a^{4} + \frac{525398757}{79925774} a^{3} - \frac{216472536}{39962887} a^{2} + \frac{218315452}{39962887} a - \frac{83240710}{39962887} \) (order $4$)
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:  \( 34612.1677943 \)
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

\(\Q(\sqrt{-1}) \), 10.2.82538991616.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.8.14.2$x^{8} + 2 x^{7} + 2$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.18.51$x^{12} + 10 x^{11} + 16 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} - 8 x^{6} - 8 x^{5} + 4 x^{4} - 8 x^{3} + 16 x + 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$