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Properties

Label 20.16.5419515012...9041.1
Degree $20$
Signature $[16, 2]$
Discriminant $71^{2}\cdot 401^{10}$
Root discriminant $30.67$
Ramified primes $71, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1087, 6964, -11666, -8367, 36358, -11884, -34682, 24951, 10351, -15189, 3096, 2880, -2794, 788, 622, -472, -20, 82, -10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 10*x^18 + 82*x^17 - 20*x^16 - 472*x^15 + 622*x^14 + 788*x^13 - 2794*x^12 + 2880*x^11 + 3096*x^10 - 15189*x^9 + 10351*x^8 + 24951*x^7 - 34682*x^6 - 11884*x^5 + 36358*x^4 - 8367*x^3 - 11666*x^2 + 6964*x - 1087)
 
gp: K = bnfinit(x^20 - 5*x^19 - 10*x^18 + 82*x^17 - 20*x^16 - 472*x^15 + 622*x^14 + 788*x^13 - 2794*x^12 + 2880*x^11 + 3096*x^10 - 15189*x^9 + 10351*x^8 + 24951*x^7 - 34682*x^6 - 11884*x^5 + 36358*x^4 - 8367*x^3 - 11666*x^2 + 6964*x - 1087, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 10 x^{18} + 82 x^{17} - 20 x^{16} - 472 x^{15} + 622 x^{14} + 788 x^{13} - 2794 x^{12} + 2880 x^{11} + 3096 x^{10} - 15189 x^{9} + 10351 x^{8} + 24951 x^{7} - 34682 x^{6} - 11884 x^{5} + 36358 x^{4} - 8367 x^{3} - 11666 x^{2} + 6964 x - 1087 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(541951501229229898847195369041=71^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.67$
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:  $71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{15} - \frac{1}{3} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{2} a^{12} + \frac{1}{6} a^{9} - \frac{1}{2} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{102} a^{18} + \frac{2}{51} a^{17} + \frac{2}{51} a^{16} - \frac{2}{51} a^{15} - \frac{4}{51} a^{14} - \frac{31}{102} a^{13} - \frac{4}{51} a^{12} + \frac{19}{51} a^{11} - \frac{49}{102} a^{10} - \frac{2}{17} a^{9} - \frac{31}{102} a^{8} + \frac{14}{51} a^{7} - \frac{10}{51} a^{6} + \frac{5}{34} a^{5} - \frac{11}{51} a^{4} - \frac{1}{51} a^{3} - \frac{5}{34} a^{2} - \frac{3}{34} a - \frac{5}{51}$, $\frac{1}{431748523683302074452402} a^{19} + \frac{756085576054477048973}{431748523683302074452402} a^{18} - \frac{21449457425116930958599}{431748523683302074452402} a^{17} + \frac{13356332994038480188481}{431748523683302074452402} a^{16} + \frac{1862531298034338119060}{215874261841651037226201} a^{15} - \frac{67976874859471185015677}{215874261841651037226201} a^{14} + \frac{434470485050971306504}{12698485990685355130953} a^{13} - \frac{106334106477188887694053}{431748523683302074452402} a^{12} + \frac{79637345263657550940629}{431748523683302074452402} a^{11} - \frac{40983853189882423995508}{215874261841651037226201} a^{10} - \frac{49085081110332178977971}{143916174561100691484134} a^{9} + \frac{100564439547088565900381}{215874261841651037226201} a^{8} - \frac{94196399781663077806366}{215874261841651037226201} a^{7} + \frac{33004806018454418575069}{215874261841651037226201} a^{6} - \frac{37793435364503685868777}{215874261841651037226201} a^{5} + \frac{162369557966970686106905}{431748523683302074452402} a^{4} - \frac{36561493728979040852785}{431748523683302074452402} a^{3} + \frac{184348141788651546076243}{431748523683302074452402} a^{2} + \frac{1546183548213828414931}{8465657327123570087302} a - \frac{38983164992863555253059}{215874261841651037226201}$

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:  $17$
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:  \( 102546280.44 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.8.736173553742071.1, 10.8.1835844273671.1, 10.10.10368641602001.1

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 32 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed