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Properties

Label 20.0.50152210390...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{23}\cdot 29^{10}$
Root discriminant $34.28$
Ramified primes $5, 29$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1646, 1395, 115, 2760, 5925, -399, -5325, 1095, 4525, -165, -2024, 30, 870, 105, -280, -24, 80, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 80*x^16 - 24*x^15 - 280*x^14 + 105*x^13 + 870*x^12 + 30*x^11 - 2024*x^10 - 165*x^9 + 4525*x^8 + 1095*x^7 - 5325*x^6 - 399*x^5 + 5925*x^4 + 2760*x^3 + 115*x^2 + 1395*x + 1646)
 
gp: K = bnfinit(x^20 - 10*x^18 + 80*x^16 - 24*x^15 - 280*x^14 + 105*x^13 + 870*x^12 + 30*x^11 - 2024*x^10 - 165*x^9 + 4525*x^8 + 1095*x^7 - 5325*x^6 - 399*x^5 + 5925*x^4 + 2760*x^3 + 115*x^2 + 1395*x + 1646, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 80 x^{16} - 24 x^{15} - 280 x^{14} + 105 x^{13} + 870 x^{12} + 30 x^{11} - 2024 x^{10} - 165 x^{9} + 4525 x^{8} + 1095 x^{7} - 5325 x^{6} - 399 x^{5} + 5925 x^{4} + 2760 x^{3} + 115 x^{2} + 1395 x + 1646 \)

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:  \(5015221039059174060821533203125=5^{23}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.28$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{66} a^{16} + \frac{1}{22} a^{15} + \frac{5}{33} a^{14} + \frac{2}{11} a^{13} - \frac{7}{66} a^{12} + \frac{5}{11} a^{11} + \frac{2}{33} a^{10} - \frac{4}{11} a^{9} - \frac{29}{66} a^{8} - \frac{7}{22} a^{7} - \frac{17}{66} a^{6} + \frac{5}{22} a^{5} - \frac{7}{33} a^{4} + \frac{1}{22} a^{3} - \frac{4}{11} a^{2} + \frac{14}{33}$, $\frac{1}{66} a^{17} + \frac{1}{66} a^{15} + \frac{5}{22} a^{14} - \frac{5}{33} a^{13} - \frac{5}{22} a^{12} - \frac{10}{33} a^{11} - \frac{1}{22} a^{10} - \frac{23}{66} a^{9} - \frac{10}{33} a^{7} - \frac{1}{2} a^{6} - \frac{13}{33} a^{5} + \frac{2}{11} a^{4} + \frac{1}{11} a^{2} - \frac{5}{66} a - \frac{3}{11}$, $\frac{1}{132} a^{18} - \frac{1}{132} a^{17} + \frac{1}{12} a^{15} + \frac{31}{132} a^{14} + \frac{4}{33} a^{13} - \frac{31}{132} a^{12} - \frac{23}{66} a^{11} + \frac{3}{44} a^{10} + \frac{47}{132} a^{9} - \frac{2}{11} a^{8} - \frac{29}{66} a^{7} - \frac{3}{44} a^{6} - \frac{5}{66} a^{5} + \frac{1}{66} a^{4} + \frac{1}{44} a^{3} + \frac{23}{66} a^{2} - \frac{13}{132} a - \frac{5}{66}$, $\frac{1}{50481971153046581895368176236} a^{19} + \frac{11247159225597175746778503}{16827323717682193965122725412} a^{18} + \frac{75419821851575095692581186}{12620492788261645473842044059} a^{17} + \frac{189076919595976699778113885}{50481971153046581895368176236} a^{16} + \frac{1461871916823731048610775607}{16827323717682193965122725412} a^{15} - \frac{387729790579913730009948743}{12620492788261645473842044059} a^{14} - \frac{3375403031490816293703677527}{16827323717682193965122725412} a^{13} - \frac{4980752899306241554496839805}{25240985576523290947684088118} a^{12} + \frac{5432696531808322154686150595}{50481971153046581895368176236} a^{11} - \frac{586572463435727544923242109}{1628450682356341351463489556} a^{10} - \frac{3865314091208631803921188069}{12620492788261645473842044059} a^{9} - \frac{8476959927193518720947456593}{25240985576523290947684088118} a^{8} + \frac{1698489433920066252600514025}{50481971153046581895368176236} a^{7} + \frac{8272381163584509488903873183}{25240985576523290947684088118} a^{6} - \frac{10969914804492480299432127007}{25240985576523290947684088118} a^{5} + \frac{2149819663591966550067391543}{50481971153046581895368176236} a^{4} - \frac{4205927643666404118920556619}{25240985576523290947684088118} a^{3} + \frac{6164277860280808191090732483}{16827323717682193965122725412} a^{2} + \frac{3082741085437462312958907161}{25240985576523290947684088118} a + \frac{5877887623896844288529537998}{12620492788261645473842044059}$

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

Class group and class number

$C_{8}$, 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:  \( 24284084.6035 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{145}) \), 4.0.105125.1, 5.1.2628125.1, 10.2.1001520947265625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$