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Properties

Label 20.0.22027144683...3744.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 13^{10}\cdot 101^{5}\cdot 347^{4}$
Root discriminant $73.65$
Ramified primes $2, 13, 101, 347$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10510100501, 0, 12383187719, 0, 5621322256, 0, 1265444251, 0, 152718262, 0, 9512779, 0, 196025, 0, -7693, 0, -67, 0, 35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 35*x^18 - 67*x^16 - 7693*x^14 + 196025*x^12 + 9512779*x^10 + 152718262*x^8 + 1265444251*x^6 + 5621322256*x^4 + 12383187719*x^2 + 10510100501)
 
gp: K = bnfinit(x^20 + 35*x^18 - 67*x^16 - 7693*x^14 + 196025*x^12 + 9512779*x^10 + 152718262*x^8 + 1265444251*x^6 + 5621322256*x^4 + 12383187719*x^2 + 10510100501, 1)
 

Normalized defining polynomial

\( x^{20} + 35 x^{18} - 67 x^{16} - 7693 x^{14} + 196025 x^{12} + 9512779 x^{10} + 152718262 x^{8} + 1265444251 x^{6} + 5621322256 x^{4} + 12383187719 x^{2} + 10510100501 \)

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:  \(22027144683070052721556702905834143744=2^{20}\cdot 13^{10}\cdot 101^{5}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.65$
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:  $2, 13, 101, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{101} a^{12} + \frac{35}{101} a^{10} + \frac{34}{101} a^{8} - \frac{17}{101} a^{6} - \frac{16}{101} a^{4} - \frac{7}{101} a^{2}$, $\frac{1}{101} a^{13} + \frac{35}{101} a^{11} + \frac{34}{101} a^{9} - \frac{17}{101} a^{7} - \frac{16}{101} a^{5} - \frac{7}{101} a^{3}$, $\frac{1}{10201} a^{14} + \frac{35}{10201} a^{12} - \frac{67}{10201} a^{10} + \frac{2508}{10201} a^{8} + \frac{2206}{10201} a^{6} - \frac{4754}{10201} a^{4} - \frac{9}{101} a^{2}$, $\frac{1}{10201} a^{15} + \frac{35}{10201} a^{13} - \frac{67}{10201} a^{11} + \frac{2508}{10201} a^{9} + \frac{2206}{10201} a^{7} - \frac{4754}{10201} a^{5} - \frac{9}{101} a^{3}$, $\frac{1}{1030301} a^{16} + \frac{35}{1030301} a^{14} - \frac{67}{1030301} a^{12} - \frac{7693}{1030301} a^{10} + \frac{196025}{1030301} a^{8} + \frac{240070}{1030301} a^{6} + \frac{2314}{10201} a^{4} + \frac{23}{101} a^{2}$, $\frac{1}{1030301} a^{17} + \frac{35}{1030301} a^{15} - \frac{67}{1030301} a^{13} - \frac{7693}{1030301} a^{11} + \frac{196025}{1030301} a^{9} + \frac{240070}{1030301} a^{7} + \frac{2314}{10201} a^{5} + \frac{23}{101} a^{3}$, $\frac{1}{11836638155988990325512733268034541} a^{18} - \frac{3302430557091140097714887284}{11836638155988990325512733268034541} a^{16} - \frac{176091677443370066771327003982}{11836638155988990325512733268034541} a^{14} - \frac{6621555538627028412540731840383}{11836638155988990325512733268034541} a^{12} - \frac{1727395869808515877799938504912121}{11836638155988990325512733268034541} a^{10} + \frac{1323739612960052493944133457948433}{11836638155988990325512733268034541} a^{8} + \frac{39049629918734212264032243675196}{117194437188009805203096368990441} a^{6} + \frac{135664510859107345990852616135}{1160340962257522823793033356341} a^{4} - \frac{3184988004107194687476518051}{11488524378787354691020132241} a^{2} + \frac{44544616109738559972595048}{113747766126607472188318141}$, $\frac{1}{11836638155988990325512733268034541} a^{19} - \frac{3302430557091140097714887284}{11836638155988990325512733268034541} a^{17} - \frac{176091677443370066771327003982}{11836638155988990325512733268034541} a^{15} - \frac{6621555538627028412540731840383}{11836638155988990325512733268034541} a^{13} - \frac{1727395869808515877799938504912121}{11836638155988990325512733268034541} a^{11} + \frac{1323739612960052493944133457948433}{11836638155988990325512733268034541} a^{9} + \frac{39049629918734212264032243675196}{117194437188009805203096368990441} a^{7} + \frac{135664510859107345990852616135}{1160340962257522823793033356341} a^{5} - \frac{3184988004107194687476518051}{11488524378787354691020132241} a^{3} + \frac{44544616109738559972595048}{113747766126607472188318141} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 4549501458.13 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101Data not computed
347Data not computed