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Properties

Label 20.20.2573949951...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{50}\cdot 3^{14}\cdot 5^{14}\cdot 23^{8}$
Root discriminant $131.99$
Ramified primes $2, 3, 5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2:Q_8$ (as 20T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-82487645, -50884740, 299329500, 293301600, -261172829, -379196064, 31646308, 179834256, 29464915, -40112652, -11471684, 4608864, 1734073, -281880, -134648, 8736, 5641, -108, -120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 120*x^18 - 108*x^17 + 5641*x^16 + 8736*x^15 - 134648*x^14 - 281880*x^13 + 1734073*x^12 + 4608864*x^11 - 11471684*x^10 - 40112652*x^9 + 29464915*x^8 + 179834256*x^7 + 31646308*x^6 - 379196064*x^5 - 261172829*x^4 + 293301600*x^3 + 299329500*x^2 - 50884740*x - 82487645)
 
gp: K = bnfinit(x^20 - 120*x^18 - 108*x^17 + 5641*x^16 + 8736*x^15 - 134648*x^14 - 281880*x^13 + 1734073*x^12 + 4608864*x^11 - 11471684*x^10 - 40112652*x^9 + 29464915*x^8 + 179834256*x^7 + 31646308*x^6 - 379196064*x^5 - 261172829*x^4 + 293301600*x^3 + 299329500*x^2 - 50884740*x - 82487645, 1)
 

Normalized defining polynomial

\( x^{20} - 120 x^{18} - 108 x^{17} + 5641 x^{16} + 8736 x^{15} - 134648 x^{14} - 281880 x^{13} + 1734073 x^{12} + 4608864 x^{11} - 11471684 x^{10} - 40112652 x^{9} + 29464915 x^{8} + 179834256 x^{7} + 31646308 x^{6} - 379196064 x^{5} - 261172829 x^{4} + 293301600 x^{3} + 299329500 x^{2} - 50884740 x - 82487645 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2573949951469829350637332070400000000000000=2^{50}\cdot 3^{14}\cdot 5^{14}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.99$
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:  $2, 3, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{90} a^{11} + \frac{1}{45} a^{10} - \frac{1}{90} a^{9} + \frac{2}{45} a^{8} + \frac{8}{45} a^{7} + \frac{2}{9} a^{6} - \frac{37}{90} a^{5} + \frac{14}{45} a^{4} + \frac{5}{18} a^{3} + \frac{2}{9} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{720} a^{12} + \frac{1}{72} a^{10} - \frac{1}{30} a^{9} + \frac{1}{90} a^{8} + \frac{7}{30} a^{7} + \frac{59}{360} a^{6} - \frac{2}{5} a^{5} - \frac{83}{360} a^{4} + \frac{1}{3} a^{3} + \frac{17}{36} a^{2} - \frac{1}{12} a - \frac{19}{144}$, $\frac{1}{2160} a^{13} - \frac{1}{2160} a^{12} + \frac{1}{216} a^{11} + \frac{43}{1080} a^{10} + \frac{2}{135} a^{9} + \frac{1}{54} a^{8} - \frac{5}{216} a^{7} - \frac{143}{1080} a^{6} + \frac{61}{1080} a^{5} + \frac{503}{1080} a^{4} - \frac{31}{108} a^{3} - \frac{25}{54} a^{2} - \frac{7}{432} a + \frac{139}{432}$, $\frac{1}{4320} a^{14} - \frac{1}{4320} a^{13} + \frac{1}{4320} a^{12} + \frac{7}{2160} a^{11} + \frac{79}{2160} a^{10} - \frac{1}{135} a^{9} + \frac{31}{432} a^{8} - \frac{79}{432} a^{7} + \frac{7}{216} a^{6} + \frac{251}{2160} a^{5} - \frac{751}{2160} a^{4} - \frac{43}{108} a^{3} - \frac{187}{864} a^{2} + \frac{103}{864} a - \frac{5}{96}$, $\frac{1}{4320} a^{15} - \frac{1}{1440} a^{12} - \frac{1}{216} a^{11} + \frac{47}{720} a^{10} + \frac{91}{2160} a^{9} + \frac{1}{90} a^{8} - \frac{133}{2160} a^{7} - \frac{47}{720} a^{6} - \frac{29}{540} a^{5} - \frac{67}{144} a^{4} - \frac{65}{288} a^{3} + \frac{19}{72} a^{2} - \frac{31}{432} a + \frac{19}{288}$, $\frac{1}{21600} a^{16} + \frac{1}{10800} a^{15} + \frac{1}{10800} a^{14} + \frac{1}{7200} a^{13} - \frac{7}{10800} a^{12} - \frac{17}{10800} a^{11} + \frac{221}{10800} a^{10} - \frac{41}{5400} a^{9} + \frac{409}{10800} a^{8} + \frac{1363}{10800} a^{7} - \frac{11}{540} a^{6} - \frac{731}{2160} a^{5} - \frac{1451}{21600} a^{4} + \frac{7}{240} a^{3} + \frac{31}{1080} a^{2} + \frac{1499}{4320} a - \frac{203}{2160}$, $\frac{1}{64800} a^{17} - \frac{1}{64800} a^{16} - \frac{1}{16200} a^{15} + \frac{7}{64800} a^{14} + \frac{7}{64800} a^{13} + \frac{1}{8100} a^{12} - \frac{89}{16200} a^{11} - \frac{53}{1296} a^{10} - \frac{61}{6480} a^{9} + \frac{763}{16200} a^{8} + \frac{3341}{32400} a^{7} - \frac{169}{6480} a^{6} - \frac{25781}{64800} a^{5} + \frac{31703}{64800} a^{4} + \frac{353}{6480} a^{3} - \frac{223}{12960} a^{2} - \frac{193}{12960} a + \frac{1799}{6480}$, $\frac{1}{2462400} a^{18} - \frac{1}{273600} a^{17} + \frac{17}{1231200} a^{16} + \frac{23}{820800} a^{15} + \frac{103}{1231200} a^{14} - \frac{131}{820800} a^{13} + \frac{7}{10944} a^{12} + \frac{283}{68400} a^{11} + \frac{10211}{123120} a^{10} - \frac{2171}{51300} a^{9} - \frac{2243}{307800} a^{8} + \frac{30947}{205200} a^{7} - \frac{2171}{14400} a^{6} + \frac{131749}{273600} a^{5} + \frac{345623}{1231200} a^{4} + \frac{16991}{54720} a^{3} - \frac{104977}{246240} a^{2} - \frac{16247}{54720} a + \frac{2779}{25920}$, $\frac{1}{393035224638748971491764488000} a^{19} - \frac{3769838061989179832383}{24564701539921810718235280500} a^{18} - \frac{2536338295504489006087961}{393035224638748971491764488000} a^{17} - \frac{61286994468876733652591}{4137212890934199699913310400} a^{16} + \frac{3596772323718881549431451}{393035224638748971491764488000} a^{15} + \frac{41362424103729488654044123}{393035224638748971491764488000} a^{14} - \frac{7016597887294256210579387}{65505870773124828581960748000} a^{13} - \frac{13169534545262923272830663}{131011741546249657163921496000} a^{12} + \frac{97835948801322094060263263}{19651761231937448574588224400} a^{11} - \frac{298745926059594608953201279}{98258806159687242872941122000} a^{10} + \frac{2794375394507111033705652431}{98258806159687242872941122000} a^{9} - \frac{1820509100513384630838093059}{24564701539921810718235280500} a^{8} + \frac{139459130343294544293029779}{1000089630124043184457416000} a^{7} - \frac{524002432338561690474982391}{3275293538656241429098037400} a^{6} - \frac{41418300349291716850802140007}{393035224638748971491764488000} a^{5} + \frac{54638137629152970337396760777}{393035224638748971491764488000} a^{4} - \frac{2742093166031849848038641179}{15721408985549958859670579520} a^{3} - \frac{6989301241993775163187773527}{15721408985549958859670579520} a^{2} + \frac{3568414128224664827502394561}{7860704492774979429835289760} a + \frac{1766033035342216672859529983}{4137212890934199699913310400}$

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:  $19$
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:  \( 7402171873190000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:Q_8$ (as 20T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 8 conjugacy class representatives for $C_5^2:Q_8$
Character table for $C_5^2:Q_8$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}, \sqrt{6})\), 10.10.320870687440896000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.22.3$x^{8} + 8 x^{6} + 6 x^{4} + 16 x^{2} + 16 x + 4$$4$$2$$22$$Q_8$$[3, 4]^{2}$
2.8.22.3$x^{8} + 8 x^{6} + 6 x^{4} + 16 x^{2} + 16 x + 4$$4$$2$$22$$Q_8$$[3, 4]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$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}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.10.8.1$x^{10} - 23 x^{5} + 3703$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$