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Properties

Label 20.12.1103422229...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 13^{2}\cdot 401^{8}$
Root discriminant $31.78$
Ramified primes $5, 13, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 3, -208, -181, 769, 487, -668, 687, -436, -3149, 372, 2827, 182, -767, -169, -38, 68, 38, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 15*x^18 + 38*x^17 + 68*x^16 - 38*x^15 - 169*x^14 - 767*x^13 + 182*x^12 + 2827*x^11 + 372*x^10 - 3149*x^9 - 436*x^8 + 687*x^7 - 668*x^6 + 487*x^5 + 769*x^4 - 181*x^3 - 208*x^2 + 3*x + 9)
 
gp: K = bnfinit(x^20 - 3*x^19 - 15*x^18 + 38*x^17 + 68*x^16 - 38*x^15 - 169*x^14 - 767*x^13 + 182*x^12 + 2827*x^11 + 372*x^10 - 3149*x^9 - 436*x^8 + 687*x^7 - 668*x^6 + 487*x^5 + 769*x^4 - 181*x^3 - 208*x^2 + 3*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 15 x^{18} + 38 x^{17} + 68 x^{16} - 38 x^{15} - 169 x^{14} - 767 x^{13} + 182 x^{12} + 2827 x^{11} + 372 x^{10} - 3149 x^{9} - 436 x^{8} + 687 x^{7} - 668 x^{6} + 487 x^{5} + 769 x^{4} - 181 x^{3} - 208 x^{2} + 3 x + 9 \)

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:  \(1103422229363422399032900390625=5^{10}\cdot 13^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.78$
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:  $5, 13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{83} a^{18} + \frac{41}{83} a^{17} - \frac{10}{83} a^{16} + \frac{41}{83} a^{15} + \frac{25}{83} a^{14} + \frac{11}{83} a^{13} - \frac{6}{83} a^{12} + \frac{13}{83} a^{11} + \frac{11}{83} a^{10} + \frac{10}{83} a^{9} + \frac{30}{83} a^{8} + \frac{18}{83} a^{7} + \frac{4}{83} a^{6} + \frac{21}{83} a^{5} + \frac{32}{83} a^{4} - \frac{28}{83} a^{3} - \frac{14}{83} a^{2} + \frac{24}{83} a - \frac{28}{83}$, $\frac{1}{858182508357139840772787} a^{19} - \frac{548050330740622099150}{286060836119046613590929} a^{18} + \frac{38213102629781511824975}{286060836119046613590929} a^{17} + \frac{123205037988517775263661}{858182508357139840772787} a^{16} + \frac{179323846727094341991599}{858182508357139840772787} a^{15} - \frac{351198115337492224161326}{858182508357139840772787} a^{14} - \frac{367175506014644316819328}{858182508357139840772787} a^{13} - \frac{29962483063360186838879}{858182508357139840772787} a^{12} - \frac{330963152820307201535398}{858182508357139840772787} a^{11} - \frac{329667841426646758125407}{858182508357139840772787} a^{10} - \frac{65036059266762354589728}{286060836119046613590929} a^{9} + \frac{329392164637167980526313}{858182508357139840772787} a^{8} + \frac{7815876913268061947581}{37312282972049558294469} a^{7} - \frac{35359309455072282757077}{286060836119046613590929} a^{6} - \frac{341523899879543289167975}{858182508357139840772787} a^{5} - \frac{368752963768331324793401}{858182508357139840772787} a^{4} - \frac{376802006810252521604513}{858182508357139840772787} a^{3} + \frac{257674438036121281863938}{858182508357139840772787} a^{2} - \frac{420557964145003157814877}{858182508357139840772787} a + \frac{16794495480002637918376}{286060836119046613590929}$

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:  \( 65945249.8589 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.6.1050439065040625.1, 10.10.80803005003125.1, 10.6.336140500813.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed