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Properties

Label 20.4.31103984373...0625.2
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 269^{7}$
Root discriminant $23.69$
Ramified primes $5, 269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109, -2359, 5879, -9042, 10200, -6733, 1856, 1579, -2620, 1308, -7, -516, 366, -16, -97, 91, -15, -20, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 8*x^18 - 20*x^17 - 15*x^16 + 91*x^15 - 97*x^14 - 16*x^13 + 366*x^12 - 516*x^11 - 7*x^10 + 1308*x^9 - 2620*x^8 + 1579*x^7 + 1856*x^6 - 6733*x^5 + 10200*x^4 - 9042*x^3 + 5879*x^2 - 2359*x + 109)
 
gp: K = bnfinit(x^20 - 3*x^19 + 8*x^18 - 20*x^17 - 15*x^16 + 91*x^15 - 97*x^14 - 16*x^13 + 366*x^12 - 516*x^11 - 7*x^10 + 1308*x^9 - 2620*x^8 + 1579*x^7 + 1856*x^6 - 6733*x^5 + 10200*x^4 - 9042*x^3 + 5879*x^2 - 2359*x + 109, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 8 x^{18} - 20 x^{17} - 15 x^{16} + 91 x^{15} - 97 x^{14} - 16 x^{13} + 366 x^{12} - 516 x^{11} - 7 x^{10} + 1308 x^{9} - 2620 x^{8} + 1579 x^{7} + 1856 x^{6} - 6733 x^{5} + 10200 x^{4} - 9042 x^{3} + 5879 x^{2} - 2359 x + 109 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3110398437339049957275390625=5^{15}\cdot 269^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.69$
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:  $5, 269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{240989369428415920656528684420777361} a^{19} + \frac{62090926597435603459490991725644429}{240989369428415920656528684420777361} a^{18} - \frac{67456393156628160564871722819846019}{240989369428415920656528684420777361} a^{17} + \frac{57704448955987651373334263004878910}{240989369428415920656528684420777361} a^{16} - \frac{83885488067366741745035500230319218}{240989369428415920656528684420777361} a^{15} + \frac{6314564063797461341486690033109236}{240989369428415920656528684420777361} a^{14} - \frac{1465627309359575976377978193806785}{240989369428415920656528684420777361} a^{13} - \frac{65581735914234985601205233656110124}{240989369428415920656528684420777361} a^{12} - \frac{101706393088914856689059858571318046}{240989369428415920656528684420777361} a^{11} - \frac{75683792002430445836206029740190104}{240989369428415920656528684420777361} a^{10} - \frac{93104916384673387649417831325892539}{240989369428415920656528684420777361} a^{9} - \frac{75584993036892558243650752766770848}{240989369428415920656528684420777361} a^{8} + \frac{5457672149328959309649975736028279}{240989369428415920656528684420777361} a^{7} + \frac{25545673899998454311786547418686951}{240989369428415920656528684420777361} a^{6} - \frac{28614500425895984397201971621776688}{240989369428415920656528684420777361} a^{5} - \frac{47983178507807526591171222245966366}{240989369428415920656528684420777361} a^{4} + \frac{69066351572392507253560357884606677}{240989369428415920656528684420777361} a^{3} + \frac{52441546828676077951890024522753718}{240989369428415920656528684420777361} a^{2} + \frac{31121646684974732081188358453490398}{240989369428415920656528684420777361} a - \frac{120053277152392105357020645595273835}{240989369428415920656528684420777361}$

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:  $11$
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:  \( 535138.88971 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

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 t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.33625.1, 10.2.5653203125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
269Data not computed