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Properties

Label 20.12.6584630745...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{28}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $49.08$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![69775, 54050, -509915, -793320, 259669, 1165256, 565752, -327670, -379411, -63890, 39897, 9002, -535, 3416, 1366, -358, -93, 46, -9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 9*x^18 + 46*x^17 - 93*x^16 - 358*x^15 + 1366*x^14 + 3416*x^13 - 535*x^12 + 9002*x^11 + 39897*x^10 - 63890*x^9 - 379411*x^8 - 327670*x^7 + 565752*x^6 + 1165256*x^5 + 259669*x^4 - 793320*x^3 - 509915*x^2 + 54050*x + 69775)
 
gp: K = bnfinit(x^20 - 4*x^19 - 9*x^18 + 46*x^17 - 93*x^16 - 358*x^15 + 1366*x^14 + 3416*x^13 - 535*x^12 + 9002*x^11 + 39897*x^10 - 63890*x^9 - 379411*x^8 - 327670*x^7 + 565752*x^6 + 1165256*x^5 + 259669*x^4 - 793320*x^3 - 509915*x^2 + 54050*x + 69775, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 9 x^{18} + 46 x^{17} - 93 x^{16} - 358 x^{15} + 1366 x^{14} + 3416 x^{13} - 535 x^{12} + 9002 x^{11} + 39897 x^{10} - 63890 x^{9} - 379411 x^{8} - 327670 x^{7} + 565752 x^{6} + 1165256 x^{5} + 259669 x^{4} - 793320 x^{3} - 509915 x^{2} + 54050 x + 69775 \)

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:  \(6584630745307392888012800000000000=2^{28}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.08$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{1872208715914155942698632540384570619198338928886245} a^{19} + \frac{29818627368687695692683822094487313015686128975876}{1872208715914155942698632540384570619198338928886245} a^{18} - \frac{33601995765984668088733739484762392526761314127534}{1872208715914155942698632540384570619198338928886245} a^{17} - \frac{24553982477069630200311808166418597809283817890721}{374441743182831188539726508076914123839667785777249} a^{16} + \frac{503164155526839743222455633170120984008233408970969}{1872208715914155942698632540384570619198338928886245} a^{15} - \frac{393193483243894543490956269611530988692874135158169}{1872208715914155942698632540384570619198338928886245} a^{14} - \frac{546254350428967644357302823157396061681439618442994}{1872208715914155942698632540384570619198338928886245} a^{13} + \frac{40751868269135346753975953960001806533995457320353}{374441743182831188539726508076914123839667785777249} a^{12} - \frac{408798088143087240308037690712185962435625096422899}{1872208715914155942698632540384570619198338928886245} a^{11} - \frac{295305795903597747764223116991751788271672499022476}{1872208715914155942698632540384570619198338928886245} a^{10} - \frac{715935040588094708196490158720032137945107662900631}{1872208715914155942698632540384570619198338928886245} a^{9} + \frac{224333073329163280921988692796734548994566525749002}{1872208715914155942698632540384570619198338928886245} a^{8} + \frac{709370461143299025758243135187619826212468258618644}{1872208715914155942698632540384570619198338928886245} a^{7} + \frac{934068678347774385152894440454320307685913238220161}{1872208715914155942698632540384570619198338928886245} a^{6} - \frac{829390410529000111405459805736668543972193581747136}{1872208715914155942698632540384570619198338928886245} a^{5} + \frac{182689136461617149992427675229556362080866538266154}{374441743182831188539726508076914123839667785777249} a^{4} + \frac{40670859823924677879178937984577768814437539356453}{374441743182831188539726508076914123839667785777249} a^{3} - \frac{193010909706815557784625823674838871463243590736059}{1872208715914155942698632540384570619198338928886245} a^{2} - \frac{144583075119326720548248005084635634472721044800249}{374441743182831188539726508076914123839667785777249} a + \frac{32473632824649709968887561582800903301371393977505}{374441743182831188539726508076914123839667785777249}$

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:  \( 5534015418.29 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed