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Properties

Label 20.0.98765518530...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 61^{7}\cdot 17939^{2}$
Root discriminant $25.10$
Ramified primes $5, 61, 17939$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3125, -3125, 15000, -4375, 18375, 2625, 10325, 1950, 3755, -620, 746, -749, 99, -249, 58, -32, 33, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 9*x^18 + 33*x^16 - 32*x^15 + 58*x^14 - 249*x^13 + 99*x^12 - 749*x^11 + 746*x^10 - 620*x^9 + 3755*x^8 + 1950*x^7 + 10325*x^6 + 2625*x^5 + 18375*x^4 - 4375*x^3 + 15000*x^2 - 3125*x + 3125)
 
gp: K = bnfinit(x^20 + 9*x^18 + 33*x^16 - 32*x^15 + 58*x^14 - 249*x^13 + 99*x^12 - 749*x^11 + 746*x^10 - 620*x^9 + 3755*x^8 + 1950*x^7 + 10325*x^6 + 2625*x^5 + 18375*x^4 - 4375*x^3 + 15000*x^2 - 3125*x + 3125, 1)
 

Normalized defining polynomial

\( x^{20} + 9 x^{18} + 33 x^{16} - 32 x^{15} + 58 x^{14} - 249 x^{13} + 99 x^{12} - 749 x^{11} + 746 x^{10} - 620 x^{9} + 3755 x^{8} + 1950 x^{7} + 10325 x^{6} + 2625 x^{5} + 18375 x^{4} - 4375 x^{3} + 15000 x^{2} - 3125 x + 3125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9876551853017526544345703125=5^{10}\cdot 61^{7}\cdot 17939^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.10$
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:  $5, 61, 17939$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{12} - \frac{7}{25} a^{10} - \frac{7}{25} a^{9} + \frac{3}{25} a^{8} - \frac{4}{25} a^{7} - \frac{6}{25} a^{6} - \frac{9}{25} a^{5} + \frac{6}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{13} - \frac{7}{25} a^{11} - \frac{7}{25} a^{10} + \frac{3}{25} a^{9} - \frac{4}{25} a^{8} - \frac{6}{25} a^{7} - \frac{9}{25} a^{6} + \frac{6}{25} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{125} a^{16} - \frac{1}{125} a^{14} - \frac{7}{125} a^{12} - \frac{32}{125} a^{11} + \frac{53}{125} a^{10} - \frac{54}{125} a^{9} + \frac{44}{125} a^{8} + \frac{16}{125} a^{7} + \frac{31}{125} a^{6} + \frac{9}{25} a^{5} - \frac{1}{25} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{125} a^{17} - \frac{1}{125} a^{15} - \frac{7}{125} a^{13} - \frac{7}{125} a^{12} + \frac{53}{125} a^{11} + \frac{46}{125} a^{10} + \frac{44}{125} a^{9} - \frac{34}{125} a^{8} - \frac{19}{125} a^{7} - \frac{1}{25} a^{6} + \frac{4}{25} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{625} a^{18} - \frac{1}{625} a^{16} - \frac{7}{625} a^{14} - \frac{32}{625} a^{13} + \frac{53}{625} a^{12} + \frac{71}{625} a^{11} + \frac{44}{625} a^{10} + \frac{141}{625} a^{9} - \frac{219}{625} a^{8} + \frac{59}{125} a^{7} + \frac{49}{125} a^{6} - \frac{2}{25} a^{5} + \frac{8}{25} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{8880325394757990072630266875} a^{19} - \frac{4473032661728162822167506}{8880325394757990072630266875} a^{18} + \frac{17875782996779203072835584}{8880325394757990072630266875} a^{17} - \frac{15744343213769748792314819}{8880325394757990072630266875} a^{16} - \frac{168990652780961311536100067}{8880325394757990072630266875} a^{15} + \frac{32204587536250969212292077}{1776065078951598014526053375} a^{14} - \frac{3101705859571523941126347}{71042603158063920581042135} a^{13} + \frac{838892237035256496698421383}{8880325394757990072630266875} a^{12} - \frac{1530483802673991574607470277}{8880325394757990072630266875} a^{11} - \frac{479244678942444320723776463}{8880325394757990072630266875} a^{10} + \frac{110279557684906745982860563}{355213015790319602905210675} a^{9} - \frac{2637599416364595525682886781}{8880325394757990072630266875} a^{8} + \frac{329081161963588427392648937}{1776065078951598014526053375} a^{7} + \frac{194811187940850121060824666}{1776065078951598014526053375} a^{6} + \frac{139736123730854051597443741}{355213015790319602905210675} a^{5} - \frac{99712681097055605186508891}{355213015790319602905210675} a^{4} - \frac{23704292053611618036756501}{71042603158063920581042135} a^{3} - \frac{18658173041964433390427812}{71042603158063920581042135} a^{2} - \frac{2665526249282511326564166}{14208520631612784116208427} a + \frac{2958797521324081730158917}{14208520631612784116208427}$

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:  $9$
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:  \( 372123.141049 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1525.1, 10.4.3419621875.1

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 24 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
61.6.3.2$x^{6} - 3721 x^{2} + 2269810$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17939Data not computed