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Properties

Label 20.16.1919766966...8125.2
Degree $20$
Signature $[16, 2]$
Discriminant $5^{15}\cdot 97^{2}\cdot 401^{8}$
Root discriminant $58.10$
Ramified primes $5, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-269469, -487098, 548219, 1203120, -517235, -1278764, 306672, 750086, -90661, -242262, -10497, 33271, 16605, 1666, -5059, -945, 705, 83, -45, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 45*x^18 + 83*x^17 + 705*x^16 - 945*x^15 - 5059*x^14 + 1666*x^13 + 16605*x^12 + 33271*x^11 - 10497*x^10 - 242262*x^9 - 90661*x^8 + 750086*x^7 + 306672*x^6 - 1278764*x^5 - 517235*x^4 + 1203120*x^3 + 548219*x^2 - 487098*x - 269469)
 
gp: K = bnfinit(x^20 - 2*x^19 - 45*x^18 + 83*x^17 + 705*x^16 - 945*x^15 - 5059*x^14 + 1666*x^13 + 16605*x^12 + 33271*x^11 - 10497*x^10 - 242262*x^9 - 90661*x^8 + 750086*x^7 + 306672*x^6 - 1278764*x^5 - 517235*x^4 + 1203120*x^3 + 548219*x^2 - 487098*x - 269469, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 45 x^{18} + 83 x^{17} + 705 x^{16} - 945 x^{15} - 5059 x^{14} + 1666 x^{13} + 16605 x^{12} + 33271 x^{11} - 10497 x^{10} - 242262 x^{9} - 90661 x^{8} + 750086 x^{7} + 306672 x^{6} - 1278764 x^{5} - 517235 x^{4} + 1203120 x^{3} + 548219 x^{2} - 487098 x - 269469 \)

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:  \(191976696673085084180853546142578125=5^{15}\cdot 97^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.10$
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:  $5, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{3} a^{11} - \frac{2}{9} a^{10} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} - \frac{4}{27} a^{14} + \frac{4}{27} a^{13} + \frac{10}{27} a^{11} + \frac{1}{3} a^{9} + \frac{11}{27} a^{8} - \frac{1}{27} a^{7} + \frac{8}{27} a^{6} - \frac{2}{9} a^{5} - \frac{11}{27} a^{4} - \frac{1}{3} a^{3} - \frac{5}{27} a^{2} + \frac{1}{3}$, $\frac{1}{1863} a^{18} + \frac{8}{621} a^{17} - \frac{98}{1863} a^{16} - \frac{83}{1863} a^{15} + \frac{34}{621} a^{14} - \frac{62}{1863} a^{13} - \frac{170}{1863} a^{12} + \frac{511}{1863} a^{11} + \frac{32}{207} a^{10} + \frac{344}{1863} a^{9} - \frac{500}{1863} a^{8} + \frac{163}{1863} a^{7} + \frac{365}{1863} a^{6} + \frac{73}{1863} a^{5} - \frac{221}{1863} a^{4} - \frac{37}{81} a^{3} - \frac{791}{1863} a^{2} - \frac{5}{23} a - \frac{80}{207}$, $\frac{1}{196621669554279868292665724528370093081} a^{19} - \frac{46469537484913126996402798307967377}{196621669554279868292665724528370093081} a^{18} + \frac{2599111703282768988290597232266711545}{196621669554279868292665724528370093081} a^{17} - \frac{5611949912681699696454900400658844685}{196621669554279868292665724528370093081} a^{16} + \frac{4796862779440199543794091464451297278}{196621669554279868292665724528370093081} a^{15} - \frac{5596767114999996864212735757241231781}{196621669554279868292665724528370093081} a^{14} - \frac{16690535497833846964505175610858357915}{196621669554279868292665724528370093081} a^{13} + \frac{6057901386224145054198679394515885304}{196621669554279868292665724528370093081} a^{12} - \frac{81742712284049664046218231172420302716}{196621669554279868292665724528370093081} a^{11} + \frac{4848466293938220682101255776516293154}{196621669554279868292665724528370093081} a^{10} - \frac{5115716571652289372153803294527079810}{21846852172697763143629524947596677009} a^{9} - \frac{18993907946763288459961564097918371555}{196621669554279868292665724528370093081} a^{8} + \frac{10680620642859331404263016090913675973}{21846852172697763143629524947596677009} a^{7} + \frac{11307586197407739590842073861461226579}{65540556518093289430888574842790031027} a^{6} + \frac{20190867426325165369748630113032136544}{196621669554279868292665724528370093081} a^{5} + \frac{66748234859883686960188519218255424331}{196621669554279868292665724528370093081} a^{4} + \frac{68567296833449966368496861104918081313}{196621669554279868292665724528370093081} a^{3} + \frac{55227567583489307843514399947512668703}{196621669554279868292665724528370093081} a^{2} - \frac{4085145245455786336765500762637263488}{21846852172697763143629524947596677009} a + \frac{722148873543288400871230100777413}{57643409426643174521449933898671971}$

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:  \( 125320707035 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

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

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed