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Properties

Label 20.12.1157544260...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 29^{5}\cdot 4903^{4}$
Root discriminant $28.39$
Ramified primes $5, 29, 4903$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -17, -6, 70, 85, -151, -232, 331, 249, -562, -226, 418, 208, -109, -90, 28, 15, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 10*x^18 + 15*x^17 + 28*x^16 - 90*x^15 - 109*x^14 + 208*x^13 + 418*x^12 - 226*x^11 - 562*x^10 + 249*x^9 + 331*x^8 - 232*x^7 - 151*x^6 + 85*x^5 + 70*x^4 - 6*x^3 - 17*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 10*x^18 + 15*x^17 + 28*x^16 - 90*x^15 - 109*x^14 + 208*x^13 + 418*x^12 - 226*x^11 - 562*x^10 + 249*x^9 + 331*x^8 - 232*x^7 - 151*x^6 + 85*x^5 + 70*x^4 - 6*x^3 - 17*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 10 x^{18} + 15 x^{17} + 28 x^{16} - 90 x^{15} - 109 x^{14} + 208 x^{13} + 418 x^{12} - 226 x^{11} - 562 x^{10} + 249 x^{9} + 331 x^{8} - 232 x^{7} - 151 x^{6} + 85 x^{5} + 70 x^{4} - 6 x^{3} - 17 x^{2} - x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(115754426025787089002626953125=5^{10}\cdot 29^{5}\cdot 4903^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.39$
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:  $5, 29, 4903$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{506071925} a^{18} + \frac{116949038}{506071925} a^{17} + \frac{149896758}{506071925} a^{16} - \frac{25327246}{101214385} a^{15} - \frac{17050179}{506071925} a^{14} - \frac{66650601}{506071925} a^{13} + \frac{161963558}{506071925} a^{12} + \frac{99367309}{506071925} a^{11} - \frac{220611643}{506071925} a^{10} + \frac{29312121}{506071925} a^{9} - \frac{14894316}{506071925} a^{8} - \frac{151513044}{506071925} a^{7} - \frac{172021686}{506071925} a^{6} + \frac{4070671}{101214385} a^{5} + \frac{236699548}{506071925} a^{4} - \frac{29154863}{506071925} a^{3} - \frac{3391084}{506071925} a^{2} + \frac{4071120}{20242877} a + \frac{19393534}{506071925}$, $\frac{1}{837549035875} a^{19} - \frac{339}{837549035875} a^{18} - \frac{116836071793}{837549035875} a^{17} + \frac{191706347429}{837549035875} a^{16} - \frac{268418503394}{837549035875} a^{15} - \frac{415132352093}{837549035875} a^{14} - \frac{43943739743}{167509807175} a^{13} - \frac{15963422757}{837549035875} a^{12} + \frac{267911724864}{837549035875} a^{11} + \frac{154688143207}{837549035875} a^{10} + \frac{183959928167}{837549035875} a^{9} - \frac{400318553362}{837549035875} a^{8} - \frac{190782886348}{837549035875} a^{7} + \frac{142583133702}{837549035875} a^{6} - \frac{359431879912}{837549035875} a^{5} + \frac{252893190016}{837549035875} a^{4} + \frac{104923553767}{837549035875} a^{3} - \frac{198405575882}{837549035875} a^{2} - \frac{354506485116}{837549035875} a + \frac{379368944082}{837549035875}$

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:  $15$
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:  \( 17846869.089 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 5.3.4903.1, 10.6.75123153125.1

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

Sibling fields

Degree 20 siblings: data not computed
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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
4903Data not computed