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Properties

Label 20.2.14672147998...9375.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $45.53$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3449, -187, -18875, -48647, -84687, -80518, -68633, -10865, -1878, 9340, 1240, -4485, -240, -285, 719, -9, 61, -10, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^18 - 10*x^17 + 61*x^16 - 9*x^15 + 719*x^14 - 285*x^13 - 240*x^12 - 4485*x^11 + 1240*x^10 + 9340*x^9 - 1878*x^8 - 10865*x^7 - 68633*x^6 - 80518*x^5 - 84687*x^4 - 48647*x^3 - 18875*x^2 - 187*x - 3449)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^18 - 10*x^17 + 61*x^16 - 9*x^15 + 719*x^14 - 285*x^13 - 240*x^12 - 4485*x^11 + 1240*x^10 + 9340*x^9 - 1878*x^8 - 10865*x^7 - 68633*x^6 - 80518*x^5 - 84687*x^4 - 48647*x^3 - 18875*x^2 - 187*x - 3449, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{18} - 10 x^{17} + 61 x^{16} - 9 x^{15} + 719 x^{14} - 285 x^{13} - 240 x^{12} - 4485 x^{11} + 1240 x^{10} + 9340 x^{9} - 1878 x^{8} - 10865 x^{7} - 68633 x^{6} - 80518 x^{5} - 84687 x^{4} - 48647 x^{3} - 18875 x^{2} - 187 x - 3449 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1467214799841538367562738193359375=-\,3^{10}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{33} a^{18} + \frac{2}{11} a^{17} + \frac{2}{33} a^{16} - \frac{14}{33} a^{15} + \frac{4}{11} a^{14} + \frac{2}{33} a^{13} - \frac{2}{33} a^{12} - \frac{16}{33} a^{11} + \frac{4}{33} a^{10} + \frac{8}{33} a^{9} - \frac{16}{33} a^{8} + \frac{7}{33} a^{7} + \frac{5}{33} a^{6} + \frac{14}{33} a^{5} - \frac{5}{33} a^{4} - \frac{5}{33} a^{3} - \frac{2}{11} a^{2} - \frac{4}{11} a + \frac{4}{33}$, $\frac{1}{810561424584273956936902489318255016034276143523717} a^{19} + \frac{942243753156920917907375746341883316413565100944}{73687402234933996085172953574386819639479649411247} a^{18} - \frac{393278708912097062219033984466241472036311019399974}{810561424584273956936902489318255016034276143523717} a^{17} + \frac{31365232480059478768652433791783301909767615725537}{270187141528091318978967496439418338678092047841239} a^{16} - \frac{355481212975099777562503472506207978315325615990501}{810561424584273956936902489318255016034276143523717} a^{15} - \frac{87372875169466039532961777419435448561300249950230}{810561424584273956936902489318255016034276143523717} a^{14} - \frac{37833979200793512323403547453487400945987176939312}{90062380509363772992989165479806112892697349280413} a^{13} + \frac{38821650068508935256506709289320787763010605875994}{90062380509363772992989165479806112892697349280413} a^{12} - \frac{127416089294182914914180202004153144038157201592329}{270187141528091318978967496439418338678092047841239} a^{11} - \frac{43570288322249615470920306211391412695912574029005}{270187141528091318978967496439418338678092047841239} a^{10} + \frac{110851084300026493836247792651093019447241039865021}{810561424584273956936902489318255016034276143523717} a^{9} - \frac{57495594601756071012289534962688916747887952638547}{270187141528091318978967496439418338678092047841239} a^{8} + \frac{48955894538454862606653440409947242885644296704388}{270187141528091318978967496439418338678092047841239} a^{7} - \frac{66934768255212626017811113558504366028721952119569}{810561424584273956936902489318255016034276143523717} a^{6} - \frac{125999633912830795840127292258193464951007458123280}{270187141528091318978967496439418338678092047841239} a^{5} + \frac{300792267798612738403170948946811032095173408085806}{810561424584273956936902489318255016034276143523717} a^{4} - \frac{5037105545077143684483431548384966930199827450487}{810561424584273956936902489318255016034276143523717} a^{3} - \frac{105789257897732322567909787237069446105506821470949}{270187141528091318978967496439418338678092047841239} a^{2} + \frac{130520823760447814302536909709804780409419793623312}{810561424584273956936902489318255016034276143523717} a + \frac{296274753677632211352449267451706968028502770133959}{810561424584273956936902489318255016034276143523717}$

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:  $10$
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:  \( 404035970.257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed