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Properties

Label 20.0.51885951292...2368.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{12}\cdot 3^{10}\cdot 13^{7}\cdot 43^{4}$
Root discriminant $13.67$
Ramified primes $2, 3, 13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 37, -95, 181, -281, 372, -419, 426, -386, 331, -278, 237, -207, 175, -134, 89, -48, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 20*x^18 - 48*x^17 + 89*x^16 - 134*x^15 + 175*x^14 - 207*x^13 + 237*x^12 - 278*x^11 + 331*x^10 - 386*x^9 + 426*x^8 - 419*x^7 + 372*x^6 - 281*x^5 + 181*x^4 - 95*x^3 + 37*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 20*x^18 - 48*x^17 + 89*x^16 - 134*x^15 + 175*x^14 - 207*x^13 + 237*x^12 - 278*x^11 + 331*x^10 - 386*x^9 + 426*x^8 - 419*x^7 + 372*x^6 - 281*x^5 + 181*x^4 - 95*x^3 + 37*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 20 x^{18} - 48 x^{17} + 89 x^{16} - 134 x^{15} + 175 x^{14} - 207 x^{13} + 237 x^{12} - 278 x^{11} + 331 x^{10} - 386 x^{9} + 426 x^{8} - 419 x^{7} + 372 x^{6} - 281 x^{5} + 181 x^{4} - 95 x^{3} + 37 x^{2} - 9 x + 1 \)

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:  \(51885951292865088442368=2^{12}\cdot 3^{10}\cdot 13^{7}\cdot 43^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.67$
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:  $2, 3, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{364701773} a^{19} - \frac{31550495}{364701773} a^{18} + \frac{114636090}{364701773} a^{17} + \frac{103068818}{364701773} a^{16} + \frac{138543366}{364701773} a^{15} - \frac{91522761}{364701773} a^{14} - \frac{58146833}{364701773} a^{13} - \frac{137279635}{364701773} a^{12} + \frac{146543403}{364701773} a^{11} - \frac{179622520}{364701773} a^{10} + \frac{173989735}{364701773} a^{9} - \frac{38580187}{364701773} a^{8} - \frac{130879882}{364701773} a^{7} + \frac{123026115}{364701773} a^{6} + \frac{11282965}{364701773} a^{5} + \frac{149294496}{364701773} a^{4} + \frac{148215689}{364701773} a^{3} + \frac{149531492}{364701773} a^{2} - \frac{75725453}{364701773} a - \frac{101693866}{364701773}$

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:  \( -\frac{2946815}{1345763} a^{19} + \frac{16634223}{1345763} a^{18} - \frac{52981942}{1345763} a^{17} + \frac{122284060}{1345763} a^{16} - \frac{217763082}{1345763} a^{15} + \frac{315377926}{1345763} a^{14} - \frac{400515028}{1345763} a^{13} + \frac{464486916}{1345763} a^{12} - \frac{530917332}{1345763} a^{11} + \frac{628361642}{1345763} a^{10} - \frac{749568837}{1345763} a^{9} + \frac{866975904}{1345763} a^{8} - \frac{941144487}{1345763} a^{7} + \frac{894743947}{1345763} a^{6} - \frac{774716693}{1345763} a^{5} + \frac{551398157}{1345763} a^{4} - \frac{339643150}{1345763} a^{3} + \frac{161467997}{1345763} a^{2} - \frac{54734262}{1345763} a + \frac{9459824}{1345763} \) (order $6$)
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:  \( 3440.36919796 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 10.0.4859704512.1

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 R R $20$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
3Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
43.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$