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Properties

Label 20.4.15691000132...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{12}\cdot 6329^{6}$
Root discriminant $36.29$
Ramified primes $5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18779, -41617, 18615, 50637, -109879, 97314, -28555, -33683, 67541, -67547, 45830, -24090, 9120, -2140, 151, 176, -60, 0, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 60*x^16 + 176*x^15 + 151*x^14 - 2140*x^13 + 9120*x^12 - 24090*x^11 + 45830*x^10 - 67547*x^9 + 67541*x^8 - 33683*x^7 - 28555*x^6 + 97314*x^5 - 109879*x^4 + 50637*x^3 + 18615*x^2 - 41617*x + 18779)
 
gp: K = bnfinit(x^20 - 5*x^19 + 11*x^18 - 60*x^16 + 176*x^15 + 151*x^14 - 2140*x^13 + 9120*x^12 - 24090*x^11 + 45830*x^10 - 67547*x^9 + 67541*x^8 - 33683*x^7 - 28555*x^6 + 97314*x^5 - 109879*x^4 + 50637*x^3 + 18615*x^2 - 41617*x + 18779, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 11 x^{18} - 60 x^{16} + 176 x^{15} + 151 x^{14} - 2140 x^{13} + 9120 x^{12} - 24090 x^{11} + 45830 x^{10} - 67547 x^{9} + 67541 x^{8} - 33683 x^{7} - 28555 x^{6} + 97314 x^{5} - 109879 x^{4} + 50637 x^{3} + 18615 x^{2} - 41617 x + 18779 \)

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:  \(15691000132788774317754150390625=5^{12}\cdot 6329^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.29$
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:  $5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{125556703325929924130494172918306179536631935025} a^{19} - \frac{6083052965112635940193252292184127486551699472}{125556703325929924130494172918306179536631935025} a^{18} - \frac{248443003127595729947675345038166465776318436}{5022268133037196965219766916732247181465277401} a^{17} + \frac{133615989394200017219580184199175315277833878}{5022268133037196965219766916732247181465277401} a^{16} - \frac{212368677478752380343975773353707020949062612}{25111340665185984826098834583661235907326387005} a^{15} - \frac{46067411791831527400791193464408767075590545154}{125556703325929924130494172918306179536631935025} a^{14} + \frac{2475341351816664612304116087731066629266465194}{125556703325929924130494172918306179536631935025} a^{13} + \frac{52919420417664964398655008518961481428583907777}{125556703325929924130494172918306179536631935025} a^{12} + \frac{46492969693990986054104281325174439982671169776}{125556703325929924130494172918306179536631935025} a^{11} + \frac{28088684673268319615818065654796851868376357428}{125556703325929924130494172918306179536631935025} a^{10} - \frac{47559448233509423299200389028861853240256340461}{125556703325929924130494172918306179536631935025} a^{9} - \frac{9938528775053549728393677131306644140620334574}{25111340665185984826098834583661235907326387005} a^{8} - \frac{31412022046506869616883210280438140707344711804}{125556703325929924130494172918306179536631935025} a^{7} - \frac{8131269714384321353795046842487566265672273347}{25111340665185984826098834583661235907326387005} a^{6} - \frac{2331309313669776990558334229835558724463636062}{25111340665185984826098834583661235907326387005} a^{5} + \frac{5386692335403262107885393220726814355749871184}{125556703325929924130494172918306179536631935025} a^{4} - \frac{34704432510909631628076375001807427300522620957}{125556703325929924130494172918306179536631935025} a^{3} + \frac{33033134901825416620978375467779216279746458216}{125556703325929924130494172918306179536631935025} a^{2} - \frac{11543954521741588513792532575026745464760564367}{125556703325929924130494172918306179536631935025} a + \frac{33693224155358384155030234858652975858673894842}{125556703325929924130494172918306179536631935025}$

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:  $11$
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:  \( 57930964.315 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ R $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
6329Data not computed