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Properties

Label 20.0.54230337251...8800.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{12}\cdot 5^{2}\cdot 17\cdot 389^{4}$
Root discriminant $17.25$
Ramified primes $2, 3, 5, 17, 389$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -5, -8, 45, -42, -73, 206, -59, -608, 1656, -2604, 2982, -2668, 1918, -1116, 522, -192, 53, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 53*x^18 - 192*x^17 + 522*x^16 - 1116*x^15 + 1918*x^14 - 2668*x^13 + 2982*x^12 - 2604*x^11 + 1656*x^10 - 608*x^9 - 59*x^8 + 206*x^7 - 73*x^6 - 42*x^5 + 45*x^4 - 8*x^3 - 5*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 53*x^18 - 192*x^17 + 522*x^16 - 1116*x^15 + 1918*x^14 - 2668*x^13 + 2982*x^12 - 2604*x^11 + 1656*x^10 - 608*x^9 - 59*x^8 + 206*x^7 - 73*x^6 - 42*x^5 + 45*x^4 - 8*x^3 - 5*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 53 x^{18} - 192 x^{17} + 522 x^{16} - 1116 x^{15} + 1918 x^{14} - 2668 x^{13} + 2982 x^{12} - 2604 x^{11} + 1656 x^{10} - 608 x^{9} - 59 x^{8} + 206 x^{7} - 73 x^{6} - 42 x^{5} + 45 x^{4} - 8 x^{3} - 5 x^{2} + 2 x + 1 \)

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:  \(5423033725190914100428800=2^{20}\cdot 3^{12}\cdot 5^{2}\cdot 17\cdot 389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.25$
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:  $2, 3, 5, 17, 389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{25} a^{18} - \frac{9}{25} a^{17} + \frac{1}{5} a^{16} - \frac{11}{25} a^{15} - \frac{9}{25} a^{14} + \frac{4}{25} a^{13} - \frac{2}{25} a^{12} - \frac{1}{25} a^{11} + \frac{9}{25} a^{10} - \frac{6}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{5} a^{6} + \frac{11}{25} a^{5} - \frac{7}{25} a^{4} - \frac{3}{25} a^{3} - \frac{2}{5} a^{2} - \frac{1}{25} a + \frac{9}{25}$, $\frac{1}{1325} a^{19} + \frac{17}{1325} a^{18} - \frac{654}{1325} a^{17} + \frac{594}{1325} a^{16} + \frac{26}{265} a^{15} + \frac{129}{265} a^{14} - \frac{648}{1325} a^{13} + \frac{347}{1325} a^{12} + \frac{108}{1325} a^{11} + \frac{153}{1325} a^{10} + \frac{593}{1325} a^{9} - \frac{126}{1325} a^{8} + \frac{71}{265} a^{7} + \frac{516}{1325} a^{6} + \frac{79}{1325} a^{5} - \frac{27}{265} a^{4} + \frac{587}{1325} a^{3} - \frac{536}{1325} a^{2} - \frac{167}{1325} a + \frac{634}{1325}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{27}{25} a^{18} + \frac{243}{25} a^{17} - \frac{232}{5} a^{16} + \frac{3772}{25} a^{15} - \frac{9107}{25} a^{14} + \frac{17017}{25} a^{13} - \frac{24896}{25} a^{12} + \frac{28177}{25} a^{11} - \frac{23493}{25} a^{10} + \frac{12162}{25} a^{9} - \frac{498}{25} a^{8} - 209 a^{7} + \frac{808}{5} a^{6} - \frac{197}{25} a^{5} - \frac{1336}{25} a^{4} + \frac{481}{25} a^{3} + \frac{49}{5} a^{2} - \frac{198}{25} a - \frac{68}{25} \) (order $4$)
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:  \( 44547.9803419 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.0.112960521216.1

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 R R $20$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
389Data not computed