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Properties

Label 20.16.7623789833...8125.2
Degree $20$
Signature $[16, 2]$
Discriminant $5^{10}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $31.20$
Ramified primes $5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -10, -231, -97, 2236, 2056, -7262, -7709, 8294, 7039, -5587, -1126, 2274, -916, -228, 256, -59, 16, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 2*x^18 + 16*x^17 - 59*x^16 + 256*x^15 - 228*x^14 - 916*x^13 + 2274*x^12 - 1126*x^11 - 5587*x^10 + 7039*x^9 + 8294*x^8 - 7709*x^7 - 7262*x^6 + 2056*x^5 + 2236*x^4 - 97*x^3 - 231*x^2 - 10*x + 5)
 
gp: K = bnfinit(x^20 - 5*x^19 + 2*x^18 + 16*x^17 - 59*x^16 + 256*x^15 - 228*x^14 - 916*x^13 + 2274*x^12 - 1126*x^11 - 5587*x^10 + 7039*x^9 + 8294*x^8 - 7709*x^7 - 7262*x^6 + 2056*x^5 + 2236*x^4 - 97*x^3 - 231*x^2 - 10*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 2 x^{18} + 16 x^{17} - 59 x^{16} + 256 x^{15} - 228 x^{14} - 916 x^{13} + 2274 x^{12} - 1126 x^{11} - 5587 x^{10} + 7039 x^{9} + 8294 x^{8} - 7709 x^{7} - 7262 x^{6} + 2056 x^{5} + 2236 x^{4} - 97 x^{3} - 231 x^{2} - 10 x + 5 \)

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:  \(762378983303206514165048828125=5^{10}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.20$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{61} a^{16} + \frac{22}{61} a^{15} + \frac{10}{61} a^{14} - \frac{26}{61} a^{13} + \frac{15}{61} a^{12} + \frac{8}{61} a^{11} + \frac{7}{61} a^{10} + \frac{30}{61} a^{9} + \frac{13}{61} a^{8} - \frac{24}{61} a^{7} - \frac{3}{61} a^{6} - \frac{16}{61} a^{5} - \frac{13}{61} a^{4} + \frac{11}{61} a^{3} - \frac{11}{61} a^{2} + \frac{23}{61} a + \frac{27}{61}$, $\frac{1}{61} a^{17} + \frac{14}{61} a^{15} - \frac{2}{61} a^{14} - \frac{23}{61} a^{13} - \frac{17}{61} a^{12} + \frac{14}{61} a^{11} - \frac{2}{61} a^{10} + \frac{24}{61} a^{9} - \frac{5}{61} a^{8} - \frac{24}{61} a^{7} - \frac{11}{61} a^{6} - \frac{27}{61} a^{5} - \frac{8}{61} a^{4} - \frac{9}{61} a^{3} + \frac{21}{61} a^{2} + \frac{9}{61} a + \frac{16}{61}$, $\frac{1}{3721} a^{18} - \frac{22}{3721} a^{17} + \frac{14}{3721} a^{16} + \frac{300}{3721} a^{15} + \frac{936}{3721} a^{14} - \frac{1463}{3721} a^{13} - \frac{1625}{3721} a^{12} - \frac{1835}{3721} a^{11} + \frac{312}{3721} a^{10} - \frac{777}{3721} a^{9} - \frac{1134}{3721} a^{8} - \frac{1252}{3721} a^{7} + \frac{1008}{3721} a^{6} + \frac{464}{3721} a^{5} - \frac{504}{3721} a^{4} - \frac{1062}{3721} a^{3} + \frac{1804}{3721} a^{2} + \frac{184}{3721} a - \frac{1511}{3721}$, $\frac{1}{23751746203709170644331} a^{19} + \frac{873924617582815615}{23751746203709170644331} a^{18} + \frac{43439156222450505163}{23751746203709170644331} a^{17} - \frac{115664211895421456601}{23751746203709170644331} a^{16} + \frac{9165257145424587395756}{23751746203709170644331} a^{15} - \frac{4827190293131047998920}{23751746203709170644331} a^{14} + \frac{4959378363765808347473}{23751746203709170644331} a^{13} - \frac{5018836743493056245488}{23751746203709170644331} a^{12} + \frac{5674983345133063689114}{23751746203709170644331} a^{11} - \frac{932018861348994805735}{23751746203709170644331} a^{10} + \frac{2358241502309029010888}{23751746203709170644331} a^{9} + \frac{7572496952537190384584}{23751746203709170644331} a^{8} - \frac{5660462748404023241728}{23751746203709170644331} a^{7} - \frac{2463641576130401547602}{23751746203709170644331} a^{6} + \frac{10523183187558420228681}{23751746203709170644331} a^{5} - \frac{603945250785872190197}{23751746203709170644331} a^{4} - \frac{827867572345501021673}{23751746203709170644331} a^{3} - \frac{5854378927760026722072}{23751746203709170644331} a^{2} - \frac{330786660732280700282}{23751746203709170644331} a + \frac{3123870353311439876380}{23751746203709170644331}$

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:  \( 156148359.668 \) (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

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61Data not computed
397Data not computed