Properties

Label 20.0.26150594311...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 199^{2}\cdot 1471^{2}$
Root discriminant $11.77$
Ramified primes $5, 199, 1471$
Class number $1$
Class group Trivial
Galois group 20T654

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 6, -6, 10, -10, 27, -48, 88, -112, 135, -128, 118, -93, 73, -50, 33, -18, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 33*x^16 - 50*x^15 + 73*x^14 - 93*x^13 + 118*x^12 - 128*x^11 + 135*x^10 - 112*x^9 + 88*x^8 - 48*x^7 + 27*x^6 - 10*x^5 + 10*x^4 - 6*x^3 + 6*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 9 x^{18} - 18 x^{17} + 33 x^{16} - 50 x^{15} + 73 x^{14} - 93 x^{13} + 118 x^{12} - 128 x^{11} + 135 x^{10} - 112 x^{9} + 88 x^{8} - 48 x^{7} + 27 x^{6} - 10 x^{5} + 10 x^{4} - 6 x^{3} + 6 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2615059431182861328125=5^{15}\cdot 199^{2}\cdot 1471^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.77$
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:  $5, 199, 1471$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{287389} a^{19} + \frac{98926}{287389} a^{18} - \frac{94743}{287389} a^{17} + \frac{74581}{287389} a^{16} + \frac{85985}{287389} a^{15} - \frac{16996}{287389} a^{14} + \frac{115828}{287389} a^{13} - \frac{26089}{287389} a^{12} + \frac{82046}{287389} a^{11} + \frac{1079}{287389} a^{10} + \frac{123207}{287389} a^{9} + \frac{2923}{287389} a^{8} + \frac{56221}{287389} a^{7} + \frac{47944}{287389} a^{6} - \frac{16053}{287389} a^{5} + \frac{4367}{287389} a^{4} + \frac{77286}{287389} a^{3} + \frac{129732}{287389} a^{2} + \frac{39072}{287389} a - \frac{28164}{287389}$

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:  \( \frac{674473}{287389} a^{19} - \frac{1912466}{287389} a^{18} + \frac{5383880}{287389} a^{17} - \frac{10321417}{287389} a^{16} + \frac{17853601}{287389} a^{15} - \frac{25835686}{287389} a^{14} + \frac{36393454}{287389} a^{13} - \frac{44617700}{287389} a^{12} + \frac{55088940}{287389} a^{11} - \frac{56815603}{287389} a^{10} + \frac{56156860}{287389} a^{9} - \frac{41100588}{287389} a^{8} + \frac{27306883}{287389} a^{7} - \frac{10997550}{287389} a^{6} + \frac{4951119}{287389} a^{5} - \frac{2612771}{287389} a^{4} + \frac{4726904}{287389} a^{3} - \frac{2609317}{287389} a^{2} + \frac{1449479}{287389} a - \frac{19450}{287389} \) (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:  \( 906.492535524 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T654:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.914778125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$199$199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
1471Data not computed