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Properties

Label 20.12.1215476281...8125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{16}\cdot 6029^{5}$
Root discriminant $31.93$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181, -1452, -7016, 10984, 20746, 4504, -8496, -14875, -8287, 5146, 5985, 1459, -480, -1152, -504, 224, 166, -14, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 21*x^18 - 14*x^17 + 166*x^16 + 224*x^15 - 504*x^14 - 1152*x^13 - 480*x^12 + 1459*x^11 + 5985*x^10 + 5146*x^9 - 8287*x^8 - 14875*x^7 - 8496*x^6 + 4504*x^5 + 20746*x^4 + 10984*x^3 - 7016*x^2 - 1452*x + 181)
 
gp: K = bnfinit(x^20 - 21*x^18 - 14*x^17 + 166*x^16 + 224*x^15 - 504*x^14 - 1152*x^13 - 480*x^12 + 1459*x^11 + 5985*x^10 + 5146*x^9 - 8287*x^8 - 14875*x^7 - 8496*x^6 + 4504*x^5 + 20746*x^4 + 10984*x^3 - 7016*x^2 - 1452*x + 181, 1)
 

Normalized defining polynomial

\( x^{20} - 21 x^{18} - 14 x^{17} + 166 x^{16} + 224 x^{15} - 504 x^{14} - 1152 x^{13} - 480 x^{12} + 1459 x^{11} + 5985 x^{10} + 5146 x^{9} - 8287 x^{8} - 14875 x^{7} - 8496 x^{6} + 4504 x^{5} + 20746 x^{4} + 10984 x^{3} - 7016 x^{2} - 1452 x + 181 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1215476281933432182159423828125=5^{16}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.93$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{32922160097064557951749195146441304531} a^{19} - \frac{9852885958231092798683195585979124707}{32922160097064557951749195146441304531} a^{18} + \frac{14505716390338497372050674107705652387}{32922160097064557951749195146441304531} a^{17} + \frac{13209588572509638172437268006971109236}{32922160097064557951749195146441304531} a^{16} + \frac{1937491743726110300087012656237258138}{32922160097064557951749195146441304531} a^{15} - \frac{11166204607718935041116813285851408159}{32922160097064557951749195146441304531} a^{14} + \frac{13162014681964405280268530626705322206}{32922160097064557951749195146441304531} a^{13} + \frac{466765591641755279658347650291304499}{32922160097064557951749195146441304531} a^{12} + \frac{16393448240236788540173056077421855134}{32922160097064557951749195146441304531} a^{11} + \frac{10530946706711748756625303758061600718}{32922160097064557951749195146441304531} a^{10} - \frac{10796160729243589437939907354170815855}{32922160097064557951749195146441304531} a^{9} - \frac{9069415091587284547117180601446098525}{32922160097064557951749195146441304531} a^{8} - \frac{14733669164747548613766713089899222991}{32922160097064557951749195146441304531} a^{7} + \frac{16043548381248472985702599796283621232}{32922160097064557951749195146441304531} a^{6} + \frac{8914594883573355385107086326096231244}{32922160097064557951749195146441304531} a^{5} - \frac{4423431447668373237005554459946906527}{32922160097064557951749195146441304531} a^{4} + \frac{126250720824640440127623252523280266}{369911911202972561255608934229677579} a^{3} - \frac{1607296856968265432990425914866747584}{32922160097064557951749195146441304531} a^{2} - \frac{6252282071494520432869838849014271065}{32922160097064557951749195146441304531} a - \frac{13920251329258221248280632081265574354}{32922160097064557951749195146441304531}$

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:  $15$
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:  \( 66918557.4571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed