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Properties

Label 20.4.29119649538...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{7}\cdot 31^{4}$
Root discriminant $42.00$
Ramified primes $2, 5, 13, 29, 31$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-281, -153, 1205, 397, -2551, -1396, 1290, 1120, 378, -712, 152, -356, -169, 621, -375, 155, -100, 37, 3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 3*x^18 + 37*x^17 - 100*x^16 + 155*x^15 - 375*x^14 + 621*x^13 - 169*x^12 - 356*x^11 + 152*x^10 - 712*x^9 + 378*x^8 + 1120*x^7 + 1290*x^6 - 1396*x^5 - 2551*x^4 + 397*x^3 + 1205*x^2 - 153*x - 281)
 
gp: K = bnfinit(x^20 - 5*x^19 + 3*x^18 + 37*x^17 - 100*x^16 + 155*x^15 - 375*x^14 + 621*x^13 - 169*x^12 - 356*x^11 + 152*x^10 - 712*x^9 + 378*x^8 + 1120*x^7 + 1290*x^6 - 1396*x^5 - 2551*x^4 + 397*x^3 + 1205*x^2 - 153*x - 281, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 3 x^{18} + 37 x^{17} - 100 x^{16} + 155 x^{15} - 375 x^{14} + 621 x^{13} - 169 x^{12} - 356 x^{11} + 152 x^{10} - 712 x^{9} + 378 x^{8} + 1120 x^{7} + 1290 x^{6} - 1396 x^{5} - 2551 x^{4} + 397 x^{3} + 1205 x^{2} - 153 x - 281 \)

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:  \(291196495384907705490560000000000=2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{7}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.00$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{146787801522031612511151521032780097} a^{19} - \frac{1155311266774500301129086435096536}{146787801522031612511151521032780097} a^{18} + \frac{1194151521648960942435681590358584}{146787801522031612511151521032780097} a^{17} + \frac{2075321953152708570122773093572228}{146787801522031612511151521032780097} a^{16} - \frac{20781462867785184122615388867518442}{146787801522031612511151521032780097} a^{15} + \frac{22935276975540562142673356113146060}{146787801522031612511151521032780097} a^{14} - \frac{51374668185687792041803812192215312}{146787801522031612511151521032780097} a^{13} + \frac{71282570545153547289628720375623586}{146787801522031612511151521032780097} a^{12} + \frac{49822639476428520462949852149600868}{146787801522031612511151521032780097} a^{11} - \frac{38857313391400225534312492559781006}{146787801522031612511151521032780097} a^{10} - \frac{63038737478943228698413899864433568}{146787801522031612511151521032780097} a^{9} - \frac{31358679707707927275089784714865161}{146787801522031612511151521032780097} a^{8} + \frac{25675107066783427076434711106715211}{146787801522031612511151521032780097} a^{7} - \frac{73110827823511341122114831826012664}{146787801522031612511151521032780097} a^{6} - \frac{29746200798338477104068997826467872}{146787801522031612511151521032780097} a^{5} - \frac{39904680018477923978102850918974568}{146787801522031612511151521032780097} a^{4} + \frac{44162248852666710001466113442219972}{146787801522031612511151521032780097} a^{3} + \frac{49437706677703129029974878718270981}{146787801522031612511151521032780097} a^{2} + \frac{51806620235731899601365146575195930}{146787801522031612511151521032780097} a - \frac{26028699171303597799569416081370474}{146787801522031612511151521032780097}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 55809266.9153 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
2Data not computed
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$