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Properties

Label 20.2.10833062044...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{38}\cdot 3^{17}\cdot 5^{15}$
Root discriminant $31.75$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-640, 2560, -5760, 9600, -11104, 10400, -7360, 5360, -3992, 3000, -1430, 400, -156, 80, 110, -100, 32, -20, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 - 20*x^17 + 32*x^16 - 100*x^15 + 110*x^14 + 80*x^13 - 156*x^12 + 400*x^11 - 1430*x^10 + 3000*x^9 - 3992*x^8 + 5360*x^7 - 7360*x^6 + 10400*x^5 - 11104*x^4 + 9600*x^3 - 5760*x^2 + 2560*x - 640)
 
gp: K = bnfinit(x^20 + 10*x^18 - 20*x^17 + 32*x^16 - 100*x^15 + 110*x^14 + 80*x^13 - 156*x^12 + 400*x^11 - 1430*x^10 + 3000*x^9 - 3992*x^8 + 5360*x^7 - 7360*x^6 + 10400*x^5 - 11104*x^4 + 9600*x^3 - 5760*x^2 + 2560*x - 640, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} - 20 x^{17} + 32 x^{16} - 100 x^{15} + 110 x^{14} + 80 x^{13} - 156 x^{12} + 400 x^{11} - 1430 x^{10} + 3000 x^{9} - 3992 x^{8} + 5360 x^{7} - 7360 x^{6} + 10400 x^{5} - 11104 x^{4} + 9600 x^{3} - 5760 x^{2} + 2560 x - 640 \)

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:  \(-1083306204463104000000000000000=-\,2^{38}\cdot 3^{17}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.75$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{144} a^{15} + \frac{1}{36} a^{14} + \frac{5}{72} a^{13} - \frac{1}{12} a^{12} + \frac{1}{9} a^{11} + \frac{5}{36} a^{10} - \frac{11}{24} a^{9} + \frac{5}{18} a^{8} + \frac{1}{4} a^{7} + \frac{4}{9} a^{6} - \frac{35}{72} a^{5} - \frac{1}{9} a^{4} + \frac{1}{18} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{288} a^{16} - \frac{1}{48} a^{14} - \frac{1}{18} a^{13} - \frac{1}{36} a^{12} + \frac{7}{72} a^{11} - \frac{1}{144} a^{10} - \frac{4}{9} a^{9} + \frac{5}{72} a^{8} + \frac{17}{36} a^{7} - \frac{19}{144} a^{6} + \frac{5}{12} a^{5} + \frac{1}{4} a^{4} + \frac{7}{36} a^{3} - \frac{1}{18} a^{2} - \frac{7}{18} a - \frac{1}{9}$, $\frac{1}{288} a^{17} + \frac{1}{36} a^{14} - \frac{5}{72} a^{13} + \frac{7}{72} a^{12} - \frac{25}{144} a^{11} - \frac{1}{36} a^{10} - \frac{11}{36} a^{9} + \frac{11}{36} a^{8} + \frac{17}{144} a^{7} + \frac{1}{4} a^{6} - \frac{5}{24} a^{5} - \frac{5}{36} a^{4} - \frac{7}{18} a^{3} + \frac{4}{9} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{576} a^{18} + \frac{5}{144} a^{14} - \frac{13}{144} a^{13} + \frac{23}{288} a^{12} + \frac{1}{72} a^{11} + \frac{5}{72} a^{10} - \frac{31}{72} a^{9} + \frac{73}{288} a^{8} - \frac{3}{8} a^{7} - \frac{71}{144} a^{6} + \frac{29}{72} a^{5} - \frac{2}{9} a^{4} - \frac{7}{18} a^{3} - \frac{5}{18} a^{2} + \frac{1}{6} a - \frac{1}{9}$, $\frac{1}{2423190627152385795523008} a^{19} - \frac{243493376157560682091}{605797656788096448880752} a^{18} + \frac{429017832425803861451}{605797656788096448880752} a^{17} + \frac{881307307106279948687}{605797656788096448880752} a^{16} - \frac{364469885242682033951}{201932552262698816293584} a^{15} + \frac{10246195197020688320555}{605797656788096448880752} a^{14} - \frac{13367434520985113554633}{134621701508465877529056} a^{13} - \frac{12959582581446628261807}{151449414197024112220188} a^{12} - \frac{10095964644563442005815}{75724707098512056110094} a^{11} + \frac{11457179265895706537677}{50483138065674704073396} a^{10} + \frac{93563341850529257042657}{1211595313576192897761504} a^{9} + \frac{14598018058139453249135}{151449414197024112220188} a^{8} + \frac{225645401316475765297127}{605797656788096448880752} a^{7} + \frac{14261806946614504967393}{50483138065674704073396} a^{6} - \frac{1027116259201096168379}{75724707098512056110094} a^{5} - \frac{16338992487823623053713}{50483138065674704073396} a^{4} + \frac{11233951268511764645825}{37862353549256028055047} a^{3} - \frac{13237696706957387711570}{37862353549256028055047} a^{2} - \frac{11332183135199818409476}{37862353549256028055047} a + \frac{14466731455622611379573}{37862353549256028055047}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 25639309.554053202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), 4.2.24000.2, 5.1.162000.1, 10.2.268738560000000.13

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed