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Properties

Label 20.12.1606601353...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{6}\cdot 31^{4}$
Root discriminant $40.77$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-164, -2112, -7788, -536, 34376, 4388, -65968, 27444, 41756, -46356, 14778, 5426, -9188, 6106, -2262, 218, 187, -104, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 36*x^18 - 104*x^17 + 187*x^16 + 218*x^15 - 2262*x^14 + 6106*x^13 - 9188*x^12 + 5426*x^11 + 14778*x^10 - 46356*x^9 + 41756*x^8 + 27444*x^7 - 65968*x^6 + 4388*x^5 + 34376*x^4 - 536*x^3 - 7788*x^2 - 2112*x - 164)
 
gp: K = bnfinit(x^20 - 9*x^19 + 36*x^18 - 104*x^17 + 187*x^16 + 218*x^15 - 2262*x^14 + 6106*x^13 - 9188*x^12 + 5426*x^11 + 14778*x^10 - 46356*x^9 + 41756*x^8 + 27444*x^7 - 65968*x^6 + 4388*x^5 + 34376*x^4 - 536*x^3 - 7788*x^2 - 2112*x - 164, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 36 x^{18} - 104 x^{17} + 187 x^{16} + 218 x^{15} - 2262 x^{14} + 6106 x^{13} - 9188 x^{12} + 5426 x^{11} + 14778 x^{10} - 46356 x^{9} + 41756 x^{8} + 27444 x^{7} - 65968 x^{6} + 4388 x^{5} + 34376 x^{4} - 536 x^{3} - 7788 x^{2} - 2112 x - 164 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(160660135384776665098240000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{6}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.77$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{20} a^{18} - \frac{1}{4} a^{17} + \frac{1}{10} a^{16} + \frac{1}{5} a^{15} - \frac{1}{4} a^{14} + \frac{1}{10} a^{13} - \frac{1}{5} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{3}{10} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{2004337947600866714960020773858460} a^{19} + \frac{18443302298804889123367931538053}{1002168973800433357480010386929230} a^{18} + \frac{228413689040409374144160256019917}{2004337947600866714960020773858460} a^{17} + \frac{25623942111462745209789436107649}{501084486900216678740005193464615} a^{16} + \frac{34994832100006632373369061987569}{2004337947600866714960020773858460} a^{15} - \frac{102823754141967414943622157853023}{2004337947600866714960020773858460} a^{14} - \frac{52933576692631440102193781240343}{501084486900216678740005193464615} a^{13} + \frac{25393229820888325203636265846062}{501084486900216678740005193464615} a^{12} + \frac{401932409555781673931482517389179}{1002168973800433357480010386929230} a^{11} - \frac{366430440170171561658113216904011}{1002168973800433357480010386929230} a^{10} + \frac{373117937042795517723958582395677}{1002168973800433357480010386929230} a^{9} + \frac{373885385491741408502610747172043}{1002168973800433357480010386929230} a^{8} - \frac{425059957540054629531702736798371}{1002168973800433357480010386929230} a^{7} + \frac{351221457325251616280208807843479}{1002168973800433357480010386929230} a^{6} - \frac{216194029980798600542789393828786}{501084486900216678740005193464615} a^{5} - \frac{148230966816074168066936773024617}{501084486900216678740005193464615} a^{4} + \frac{107838380931345663892279534051472}{501084486900216678740005193464615} a^{3} + \frac{249230912968351670821782469083256}{501084486900216678740005193464615} a^{2} + \frac{45488089416713848329404450985414}{100216897380043335748001038692923} a - \frac{97267643246025649702551210333539}{501084486900216678740005193464615}$

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:  $15$
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:  \( 919404538.339 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 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 sibling: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
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$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
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$$[\ ]_{3}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$