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Properties

Label 20.4.18660339843...5168.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{15}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $18.35$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, -7, -29, -56, -181, -159, -163, -132, 81, -86, 15, -13, -34, 63, -33, -13, 21, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 21*x^17 - 13*x^16 - 33*x^15 + 63*x^14 - 34*x^13 - 13*x^12 + 15*x^11 - 86*x^10 + 81*x^9 - 132*x^8 - 163*x^7 - 159*x^6 - 181*x^5 - 56*x^4 - 29*x^3 - 7*x^2 - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 5*x^18 + 21*x^17 - 13*x^16 - 33*x^15 + 63*x^14 - 34*x^13 - 13*x^12 + 15*x^11 - 86*x^10 + 81*x^9 - 132*x^8 - 163*x^7 - 159*x^6 - 181*x^5 - 56*x^4 - 29*x^3 - 7*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 5 x^{18} + 21 x^{17} - 13 x^{16} - 33 x^{15} + 63 x^{14} - 34 x^{13} - 13 x^{12} + 15 x^{11} - 86 x^{10} + 81 x^{9} - 132 x^{8} - 163 x^{7} - 159 x^{6} - 181 x^{5} - 56 x^{4} - 29 x^{3} - 7 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18660339843459071567495168=2^{15}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.35$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{16} + \frac{3}{14} a^{15} - \frac{1}{7} a^{14} - \frac{1}{14} a^{13} + \frac{2}{7} a^{12} + \frac{1}{7} a^{11} - \frac{5}{14} a^{10} + \frac{3}{14} a^{9} + \frac{3}{14} a^{8} - \frac{1}{14} a^{7} - \frac{1}{14} a^{5} + \frac{1}{7} a^{4} + \frac{3}{14} a^{3} - \frac{3}{14} a^{2} - \frac{5}{14} a + \frac{5}{14}$, $\frac{1}{14} a^{17} + \frac{3}{14} a^{15} + \frac{5}{14} a^{14} - \frac{1}{2} a^{13} + \frac{2}{7} a^{12} + \frac{3}{14} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{14} a^{7} - \frac{1}{14} a^{6} + \frac{5}{14} a^{5} - \frac{3}{14} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{1}{14}$, $\frac{1}{14} a^{18} + \frac{3}{14} a^{15} - \frac{1}{14} a^{14} - \frac{1}{2} a^{13} - \frac{1}{7} a^{12} + \frac{5}{14} a^{11} + \frac{1}{7} a^{10} - \frac{5}{14} a^{9} + \frac{1}{14} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{2} a^{5} - \frac{2}{7} a^{4} - \frac{5}{14} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7}$, $\frac{1}{664400904848403058} a^{19} - \frac{8952173406141135}{664400904848403058} a^{18} - \frac{10682147440438027}{664400904848403058} a^{17} - \frac{5064079058396597}{332200452424201529} a^{16} + \frac{45394739009922615}{332200452424201529} a^{15} + \frac{34463980616859699}{94914414978343294} a^{14} + \frac{148445083578778507}{664400904848403058} a^{13} + \frac{49266102017064799}{664400904848403058} a^{12} + \frac{155807487127957}{469873341476947} a^{11} + \frac{85862243208728559}{332200452424201529} a^{10} - \frac{110779738523772309}{664400904848403058} a^{9} - \frac{96814138925559677}{332200452424201529} a^{8} + \frac{93479167593268287}{332200452424201529} a^{7} - \frac{152806432714533858}{332200452424201529} a^{6} - \frac{295654450643875467}{664400904848403058} a^{5} - \frac{8828711140888977}{332200452424201529} a^{4} - \frac{66447748301961078}{332200452424201529} a^{3} + \frac{34218747619334105}{664400904848403058} a^{2} + \frac{227579845499805173}{664400904848403058} a + \frac{91744013435778378}{332200452424201529}$

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:  $11$
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:  \( 41687.7704224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 5.1.14161.1, 10.2.3409076657.1

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 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 R $20$ $20$ R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.4$x^{10} - 18 x^{8} + 88 x^{6} - 368 x^{4} + 144 x^{2} - 288$$2$$5$$15$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 3]^{5}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$