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Properties

Label 20.0.95777233176...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{12}$
Root discriminant $14.09$
Ramified primes $5, 11$
Class number $1$
Class group Trivial
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 46, -108, 101, 108, -325, 44, 749, -1227, 768, 136, -595, 403, 14, -252, 244, -134, 47, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 47*x^18 - 134*x^17 + 244*x^16 - 252*x^15 + 14*x^14 + 403*x^13 - 595*x^12 + 136*x^11 + 768*x^10 - 1227*x^9 + 749*x^8 + 44*x^7 - 325*x^6 + 108*x^5 + 101*x^4 - 108*x^3 + 46*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 47*x^18 - 134*x^17 + 244*x^16 - 252*x^15 + 14*x^14 + 403*x^13 - 595*x^12 + 136*x^11 + 768*x^10 - 1227*x^9 + 749*x^8 + 44*x^7 - 325*x^6 + 108*x^5 + 101*x^4 - 108*x^3 + 46*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 47 x^{18} - 134 x^{17} + 244 x^{16} - 252 x^{15} + 14 x^{14} + 403 x^{13} - 595 x^{12} + 136 x^{11} + 768 x^{10} - 1227 x^{9} + 749 x^{8} + 44 x^{7} - 325 x^{6} + 108 x^{5} + 101 x^{4} - 108 x^{3} + 46 x^{2} - 10 x + 1 \)

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:  \(95777233176300048828125=5^{15}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.09$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{13} + \frac{5}{11} a^{12} - \frac{3}{11} a^{10} - \frac{5}{11} a^{8} + \frac{1}{11} a^{7} + \frac{3}{11} a^{6} - \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} + \frac{5}{11} a^{13} - \frac{3}{11} a^{11} - \frac{5}{11} a^{9} + \frac{1}{11} a^{8} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} + \frac{4}{11} a^{5} - \frac{2}{11} a^{2} + \frac{1}{11} a$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{14} - \frac{5}{11} a^{13} - \frac{5}{11} a^{12} + \frac{5}{11} a^{10} + \frac{1}{11} a^{9} + \frac{5}{11} a^{8} + \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{2}{11} a^{3} + \frac{1}{11} a^{2} + \frac{3}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{14} + \frac{4}{11} a^{13} - \frac{3}{11} a^{12} + \frac{5}{11} a^{11} + \frac{5}{11} a^{10} + \frac{5}{11} a^{9} + \frac{5}{11} a^{8} - \frac{1}{11} a^{6} + \frac{4}{11} a^{5} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{3}{11} a - \frac{5}{11}$, $\frac{1}{36033777461} a^{19} - \frac{416334066}{36033777461} a^{18} + \frac{815584958}{36033777461} a^{17} + \frac{1079724971}{36033777461} a^{16} + \frac{1154595515}{36033777461} a^{15} - \frac{15706689208}{36033777461} a^{14} + \frac{3698202506}{36033777461} a^{13} + \frac{11167085334}{36033777461} a^{12} - \frac{15397561822}{36033777461} a^{11} - \frac{540407523}{36033777461} a^{10} - \frac{8893608366}{36033777461} a^{9} + \frac{5996158226}{36033777461} a^{8} + \frac{127235556}{878872621} a^{7} - \frac{14540911413}{36033777461} a^{6} + \frac{10982332388}{36033777461} a^{5} - \frac{1823914368}{36033777461} a^{4} + \frac{14540034857}{36033777461} a^{3} + \frac{11792312817}{36033777461} a^{2} - \frac{11713136756}{36033777461} a + \frac{1065438470}{36033777461}$

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:  \( \frac{2080954354}{878872621} a^{19} - \frac{20530453129}{878872621} a^{18} + \frac{94757068684}{878872621} a^{17} - \frac{23945131881}{79897511} a^{16} + \frac{41865676087}{79897511} a^{15} - \frac{431854944850}{878872621} a^{14} - \frac{77330010775}{878872621} a^{13} + \frac{863618776193}{878872621} a^{12} - \frac{1093768275598}{878872621} a^{11} + \frac{31688466847}{878872621} a^{10} + \frac{1699822099519}{878872621} a^{9} - \frac{2279787922366}{878872621} a^{8} + \frac{1041911224736}{878872621} a^{7} + \frac{437966576813}{878872621} a^{6} - \frac{659196102755}{878872621} a^{5} + \frac{60893123256}{878872621} a^{4} + \frac{267862939808}{878872621} a^{3} - \frac{173490441705}{878872621} a^{2} + \frac{44290319787}{878872621} a - \frac{3871335134}{878872621} \) (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:  \( 7329.26903421 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

Intermediate fields

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

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

Sibling fields

Degree 25 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 $20$ $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$