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Properties

Label 20.0.26228105183...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 31^{10}$
Root discriminant $37.23$
Ramified primes $2, 5, 31$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9305, -15410, 52065, -55080, 97016, -71516, 87619, -49766, 51895, -23016, 19612, -4888, 4643, -1078, 807, 128, 84, -12, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 7*x^18 - 12*x^17 + 84*x^16 + 128*x^15 + 807*x^14 - 1078*x^13 + 4643*x^12 - 4888*x^11 + 19612*x^10 - 23016*x^9 + 51895*x^8 - 49766*x^7 + 87619*x^6 - 71516*x^5 + 97016*x^4 - 55080*x^3 + 52065*x^2 - 15410*x + 9305)
 
gp: K = bnfinit(x^20 - 2*x^19 - 7*x^18 - 12*x^17 + 84*x^16 + 128*x^15 + 807*x^14 - 1078*x^13 + 4643*x^12 - 4888*x^11 + 19612*x^10 - 23016*x^9 + 51895*x^8 - 49766*x^7 + 87619*x^6 - 71516*x^5 + 97016*x^4 - 55080*x^3 + 52065*x^2 - 15410*x + 9305, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 7 x^{18} - 12 x^{17} + 84 x^{16} + 128 x^{15} + 807 x^{14} - 1078 x^{13} + 4643 x^{12} - 4888 x^{11} + 19612 x^{10} - 23016 x^{9} + 51895 x^{8} - 49766 x^{7} + 87619 x^{6} - 71516 x^{5} + 97016 x^{4} - 55080 x^{3} + 52065 x^{2} - 15410 x + 9305 \)

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:  \(26228105183385632000000000000000=2^{20}\cdot 5^{15}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.23$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{8} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{7}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{16}$, $\frac{1}{80} a^{16} + \frac{1}{40} a^{15} - \frac{1}{16} a^{13} + \frac{3}{40} a^{12} - \frac{1}{8} a^{11} - \frac{7}{40} a^{10} + \frac{3}{40} a^{8} - \frac{1}{8} a^{7} - \frac{11}{40} a^{5} + \frac{21}{80} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{3}{16} a - \frac{1}{8}$, $\frac{1}{80} a^{17} + \frac{1}{80} a^{15} - \frac{1}{16} a^{14} - \frac{1}{20} a^{13} - \frac{7}{80} a^{12} + \frac{3}{40} a^{11} + \frac{9}{40} a^{10} - \frac{1}{20} a^{9} - \frac{1}{40} a^{8} + \frac{1}{8} a^{7} - \frac{3}{20} a^{6} - \frac{7}{16} a^{5} - \frac{11}{40} a^{4} - \frac{1}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{8} a - \frac{1}{16}$, $\frac{1}{297600} a^{18} + \frac{7}{1488} a^{17} + \frac{47}{18600} a^{16} + \frac{1753}{74400} a^{15} - \frac{907}{24800} a^{14} + \frac{389}{24800} a^{13} + \frac{8329}{99200} a^{12} + \frac{16607}{74400} a^{11} + \frac{71}{300} a^{10} - \frac{14713}{74400} a^{9} - \frac{1151}{74400} a^{8} - \frac{3981}{24800} a^{7} - \frac{8177}{59520} a^{6} + \frac{3643}{24800} a^{5} - \frac{271}{12400} a^{4} + \frac{2251}{7440} a^{3} + \frac{119}{1240} a^{2} - \frac{505}{1488} a - \frac{20539}{59520}$, $\frac{1}{569429555908287505208657253604569600} a^{19} + \frac{123122682521376407722311737837}{189809851969429168402885751201523200} a^{18} - \frac{395011938711131975393235365875561}{71178694488535938151082156700571200} a^{17} - \frac{41445355786157513339404574014463}{47452462992357292100721437800380800} a^{16} - \frac{142665445759385260814764807951321}{17794673622133984537770539175142800} a^{15} + \frac{287965123746359383301714872932289}{5931557874044661512590179725047600} a^{14} + \frac{2342733571345342357279680893321833}{37961970393885833680577150240304640} a^{13} - \frac{1551086059914938249583046499362831}{22777182236331500208346290144182784} a^{12} + \frac{413404836956518619709989257576535}{1898098519694291684028857512015232} a^{11} + \frac{1022632433424318970925379876899249}{9490492598471458420144287560076160} a^{10} + \frac{4642698394457597495793674709234941}{23726231496178646050360718900190400} a^{9} + \frac{7840067812110778759197978432293989}{35589347244267969075541078350285600} a^{8} + \frac{114558055156662656448782847089526023}{569429555908287505208657253604569600} a^{7} - \frac{78575658581588755246775002733228839}{569429555908287505208657253604569600} a^{6} + \frac{16885852231429148564981338410228371}{47452462992357292100721437800380800} a^{5} - \frac{28357829049455655386785081332881983}{71178694488535938151082156700571200} a^{4} - \frac{156152098249322500692979598266801}{711786944885359381510821567005712} a^{3} + \frac{6809194727221710938258102031328579}{14235738897707187630216431340114240} a^{2} - \frac{13799832969088766563306750320549659}{113885911181657501041731450720913920} a + \frac{2738059925886554251674879112046051}{113885911181657501041731450720913920}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 30783633.4882 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1922000.2, 5.1.1922000.1 x5, 10.2.18470420000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1922000.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$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}$
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$