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Properties

Label 20.0.48828125000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{27}$
Root discriminant $15.29$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_4\times D_5$ (as 20T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 5, -5, 0, 11, 45, -170, 220, -85, -194, 350, -205, -140, 420, -464, 325, -155, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 155 x^{17} + 325 x^{16} - 464 x^{15} + 420 x^{14} - 140 x^{13} - 205 x^{12} + 350 x^{11} - 194 x^{10} - 85 x^{9} + 220 x^{8} - 170 x^{7} + 45 x^{6} + 11 x^{5} - 5 x^{3} + 5 x^{2} + 5 x + 1 \)

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:  \(488281250000000000000000=2^{16}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.29$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{48314316844849} a^{19} + \frac{2158374985593}{4392210622259} a^{18} - \frac{19312124365044}{48314316844849} a^{17} - \frac{13485989362534}{48314316844849} a^{16} + \frac{10942425054607}{48314316844849} a^{15} + \frac{920244885600}{48314316844849} a^{14} + \frac{12279019176628}{48314316844849} a^{13} - \frac{15054827396546}{48314316844849} a^{12} - \frac{22244290514076}{48314316844849} a^{11} - \frac{17706819140404}{48314316844849} a^{10} + \frac{17861746782486}{48314316844849} a^{9} + \frac{6680134690978}{48314316844849} a^{8} + \frac{18525576144657}{48314316844849} a^{7} + \frac{14134580911150}{48314316844849} a^{6} - \frac{4628551523965}{48314316844849} a^{5} + \frac{20232909133388}{48314316844849} a^{4} - \frac{9677543067033}{48314316844849} a^{3} + \frac{5178069613806}{48314316844849} a^{2} - \frac{17835699589165}{48314316844849} a + \frac{6894647134673}{48314316844849}$

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:  \( \frac{25647672629082}{48314316844849} a^{19} - \frac{23382991792054}{4392210622259} a^{18} + \frac{1293488166942520}{48314316844849} a^{17} - \frac{4046355370809716}{48314316844849} a^{16} + \frac{8607879987798252}{48314316844849} a^{15} - \frac{12590356237221986}{48314316844849} a^{14} + \frac{11974151847780594}{48314316844849} a^{13} - \frac{4978800801360226}{48314316844849} a^{12} - \frac{4386127065295506}{48314316844849} a^{11} + \frac{9176307093212860}{48314316844849} a^{10} - \frac{5968565866786384}{48314316844849} a^{9} - \frac{1317834458946326}{48314316844849} a^{8} + \frac{5659642664031292}{48314316844849} a^{7} - \frac{5007501121126595}{48314316844849} a^{6} + \frac{1722849221016958}{48314316844849} a^{5} + \frac{106128301747313}{48314316844849} a^{4} - \frac{102704872821264}{48314316844849} a^{3} - \frac{140755295603881}{48314316844849} a^{2} + \frac{185038799661752}{48314316844849} a + \frac{103109586667261}{48314316844849} \) (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:  \( 22883.422779 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_5$ (as 20T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.250000.1, 10.2.312500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 $20$ R $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed