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Properties

Label 20.12.1245907905...1889.2
Degree $20$
Signature $[12, 4]$
Discriminant $17^{2}\cdot 401^{11}$
Root discriminant $35.87$
Ramified primes $17, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![421, -2288, 2662, 8325, -26842, 19432, 23616, -45061, 10279, 25118, -17375, -2087, 5056, -1333, 143, 10, -149, 82, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 3*x^18 + 82*x^17 - 149*x^16 + 10*x^15 + 143*x^14 - 1333*x^13 + 5056*x^12 - 2087*x^11 - 17375*x^10 + 25118*x^9 + 10279*x^8 - 45061*x^7 + 23616*x^6 + 19432*x^5 - 26842*x^4 + 8325*x^3 + 2662*x^2 - 2288*x + 421)
 
gp: K = bnfinit(x^20 - 6*x^19 - 3*x^18 + 82*x^17 - 149*x^16 + 10*x^15 + 143*x^14 - 1333*x^13 + 5056*x^12 - 2087*x^11 - 17375*x^10 + 25118*x^9 + 10279*x^8 - 45061*x^7 + 23616*x^6 + 19432*x^5 - 26842*x^4 + 8325*x^3 + 2662*x^2 - 2288*x + 421, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 3 x^{18} + 82 x^{17} - 149 x^{16} + 10 x^{15} + 143 x^{14} - 1333 x^{13} + 5056 x^{12} - 2087 x^{11} - 17375 x^{10} + 25118 x^{9} + 10279 x^{8} - 45061 x^{7} + 23616 x^{6} + 19432 x^{5} - 26842 x^{4} + 8325 x^{3} + 2662 x^{2} - 2288 x + 421 \)

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:  \(12459079056924067396846384471889=17^{2}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.87$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{69} a^{18} - \frac{8}{69} a^{17} + \frac{22}{69} a^{16} - \frac{11}{69} a^{15} - \frac{7}{23} a^{14} + \frac{22}{69} a^{13} + \frac{2}{69} a^{12} - \frac{4}{23} a^{11} - \frac{31}{69} a^{10} + \frac{29}{69} a^{9} - \frac{25}{69} a^{8} - \frac{32}{69} a^{7} + \frac{7}{23} a^{6} + \frac{34}{69} a^{5} - \frac{22}{69} a^{4} + \frac{25}{69} a^{3} + \frac{9}{23} a^{2} - \frac{14}{69} a - \frac{34}{69}$, $\frac{1}{6441108488137633800627297} a^{19} - \frac{893269400522737879126}{6441108488137633800627297} a^{18} - \frac{87130622902775837727509}{6441108488137633800627297} a^{17} - \frac{711210551087116288091074}{2147036162712544600209099} a^{16} + \frac{3002227169162431131041515}{6441108488137633800627297} a^{15} - \frac{33996669329661988842235}{238559573634727177801011} a^{14} + \frac{458864865619752478966271}{6441108488137633800627297} a^{13} - \frac{710600840167189569756140}{2147036162712544600209099} a^{12} - \frac{98319253783648844303776}{280048195136418860896839} a^{11} - \frac{815239915391302074976511}{2147036162712544600209099} a^{10} - \frac{1827228321649525298156495}{6441108488137633800627297} a^{9} + \frac{22905727406165849555519}{280048195136418860896839} a^{8} - \frac{278020649823177172790237}{2147036162712544600209099} a^{7} + \frac{1350764296155679851507671}{6441108488137633800627297} a^{6} - \frac{2151822956155344120724970}{6441108488137633800627297} a^{5} - \frac{903393560164813402007015}{2147036162712544600209099} a^{4} - \frac{418704925146497248482403}{6441108488137633800627297} a^{3} - \frac{852853589012434117574024}{6441108488137633800627297} a^{2} + \frac{72073661178088923168061}{2147036162712544600209099} a + \frac{1124870308127184859203856}{6441108488137633800627297}$

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:  \( 218391620.691 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed