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Properties

Label 20.16.1543943901...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{6}\cdot 31^{6}$
Root discriminant $57.47$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-172880, -474920, 469316, 1420982, -477021, -1152224, 108136, 314242, 272542, -12424, -228826, -6726, 75572, 964, -12768, -38, 1162, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 54*x^18 + 1162*x^16 - 38*x^15 - 12768*x^14 + 964*x^13 + 75572*x^12 - 6726*x^11 - 228826*x^10 - 12424*x^9 + 272542*x^8 + 314242*x^7 + 108136*x^6 - 1152224*x^5 - 477021*x^4 + 1420982*x^3 + 469316*x^2 - 474920*x - 172880)
 
gp: K = bnfinit(x^20 - 54*x^18 + 1162*x^16 - 38*x^15 - 12768*x^14 + 964*x^13 + 75572*x^12 - 6726*x^11 - 228826*x^10 - 12424*x^9 + 272542*x^8 + 314242*x^7 + 108136*x^6 - 1152224*x^5 - 477021*x^4 + 1420982*x^3 + 469316*x^2 - 474920*x - 172880, 1)
 

Normalized defining polynomial

\( x^{20} - 54 x^{18} + 1162 x^{16} - 38 x^{15} - 12768 x^{14} + 964 x^{13} + 75572 x^{12} - 6726 x^{11} - 228826 x^{10} - 12424 x^{9} + 272542 x^{8} + 314242 x^{7} + 108136 x^{6} - 1152224 x^{5} - 477021 x^{4} + 1420982 x^{3} + 469316 x^{2} - 474920 x - 172880 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(154394390104770375159408640000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{6}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{2}$, $\frac{1}{2647084407707608645865939630247706518439063869704} a^{19} + \frac{113568610459438223101898422632507822202742443935}{1323542203853804322932969815123853259219531934852} a^{18} - \frac{246029859850273595116059405968820200493090306945}{1323542203853804322932969815123853259219531934852} a^{17} + \frac{56476040991577745044529829448884019899104544032}{330885550963451080733242453780963314804882983713} a^{16} + \frac{186870499551366791403322837380799742428729638315}{1323542203853804322932969815123853259219531934852} a^{15} - \frac{141724556905305524336242035700843621721808702439}{1323542203853804322932969815123853259219531934852} a^{14} - \frac{74830435291000353992556441309625981534685571569}{661771101926902161466484907561926629609765967426} a^{13} - \frac{77903609608286292047996572134578040770866970091}{661771101926902161466484907561926629609765967426} a^{12} + \frac{86619264931675251371378054915719358933902797571}{661771101926902161466484907561926629609765967426} a^{11} - \frac{55480719200294855169986154981958366986265119061}{1323542203853804322932969815123853259219531934852} a^{10} + \frac{71475320411271460287126834677141112572269432599}{1323542203853804322932969815123853259219531934852} a^{9} - \frac{68357036883920953994676009919030643186455395677}{330885550963451080733242453780963314804882983713} a^{8} - \frac{271325767599824238843469497074712190440999284967}{1323542203853804322932969815123853259219531934852} a^{7} + \frac{415496102626560206431547629914338692243124032117}{1323542203853804322932969815123853259219531934852} a^{6} + \frac{27023888067345610886200968743210084242057700227}{661771101926902161466484907561926629609765967426} a^{5} - \frac{16182871983834129555816207295744960756927539419}{330885550963451080733242453780963314804882983713} a^{4} + \frac{563946079486578730981676132273864344440358165195}{2647084407707608645865939630247706518439063869704} a^{3} + \frac{53435389032980754428173822993385814307269045361}{661771101926902161466484907561926629609765967426} a^{2} - \frac{53122522346146722542238687249085786608604670902}{330885550963451080733242453780963314804882983713} a - \frac{85082383181147661662244529545036288454995803677}{330885550963451080733242453780963314804882983713}$

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:  $17$
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:  \( 108830869988 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 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 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
31Data not computed