↑

Properties

Label 20.0.52409367847...3369.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{13}\cdot 19^{15}$
Root discriminant $43.25$
Ramified primes $11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![470981, 1125311, -390130, -1606905, 637422, 640998, -232705, -306777, 143855, 71697, -31273, -19300, 4811, 3987, -469, -515, -40, 77, 2, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 2*x^18 + 77*x^17 - 40*x^16 - 515*x^15 - 469*x^14 + 3987*x^13 + 4811*x^12 - 19300*x^11 - 31273*x^10 + 71697*x^9 + 143855*x^8 - 306777*x^7 - 232705*x^6 + 640998*x^5 + 637422*x^4 - 1606905*x^3 - 390130*x^2 + 1125311*x + 470981)
 
gp: K = bnfinit(x^20 - 7*x^19 + 2*x^18 + 77*x^17 - 40*x^16 - 515*x^15 - 469*x^14 + 3987*x^13 + 4811*x^12 - 19300*x^11 - 31273*x^10 + 71697*x^9 + 143855*x^8 - 306777*x^7 - 232705*x^6 + 640998*x^5 + 637422*x^4 - 1606905*x^3 - 390130*x^2 + 1125311*x + 470981, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 2 x^{18} + 77 x^{17} - 40 x^{16} - 515 x^{15} - 469 x^{14} + 3987 x^{13} + 4811 x^{12} - 19300 x^{11} - 31273 x^{10} + 71697 x^{9} + 143855 x^{8} - 306777 x^{7} - 232705 x^{6} + 640998 x^{5} + 637422 x^{4} - 1606905 x^{3} - 390130 x^{2} + 1125311 x + 470981 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(524093678472817852271376481973369=11^{13}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.25$
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:  $11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{119} a^{17} - \frac{44}{119} a^{16} - \frac{1}{17} a^{15} - \frac{6}{17} a^{14} - \frac{2}{119} a^{13} + \frac{4}{17} a^{12} - \frac{9}{119} a^{11} + \frac{4}{119} a^{10} + \frac{2}{7} a^{9} + \frac{36}{119} a^{8} + \frac{57}{119} a^{7} + \frac{2}{17} a^{6} + \frac{8}{119} a^{5} + \frac{12}{119} a^{4} + \frac{43}{119} a^{3} + \frac{55}{119} a^{2} + \frac{50}{119} a - \frac{1}{17}$, $\frac{1}{119} a^{18} - \frac{39}{119} a^{16} + \frac{1}{17} a^{15} + \frac{54}{119} a^{14} + \frac{59}{119} a^{13} + \frac{33}{119} a^{12} - \frac{5}{17} a^{11} - \frac{4}{17} a^{10} - \frac{15}{119} a^{9} - \frac{25}{119} a^{8} + \frac{23}{119} a^{7} + \frac{29}{119} a^{6} + \frac{1}{17} a^{5} - \frac{24}{119} a^{4} + \frac{43}{119} a^{3} - \frac{29}{119} a^{2} + \frac{3}{7} a + \frac{7}{17}$, $\frac{1}{18191459846216876035685364855544108881375167587159637} a^{19} - \frac{67365679941695391348596579839933059964978185766}{18191459846216876035685364855544108881375167587159637} a^{18} + \frac{69839707630711401881420565734998964018522987531246}{18191459846216876035685364855544108881375167587159637} a^{17} + \frac{472885520000765444348538561479196438751487730140122}{1070085873306875060922668520914359345963245152185861} a^{16} + \frac{3167937856567180420513755882018919756585541458846358}{18191459846216876035685364855544108881375167587159637} a^{15} - \frac{864393880865804305895147939480828289148915563462771}{2598779978030982290812194979363444125910738226737091} a^{14} + \frac{2999597776413251908833964732660170839484640118257661}{18191459846216876035685364855544108881375167587159637} a^{13} + \frac{9044174658701086266882610963385165862798961774719195}{18191459846216876035685364855544108881375167587159637} a^{12} + \frac{8370474545108522322635379201904100528232453953055720}{18191459846216876035685364855544108881375167587159637} a^{11} + \frac{6668342603893557333709009609569319183272998573488944}{18191459846216876035685364855544108881375167587159637} a^{10} - \frac{1266666808087968454922543225744123421566853855880797}{2598779978030982290812194979363444125910738226737091} a^{9} + \frac{2741077403618067011043901671639342853307019979438944}{18191459846216876035685364855544108881375167587159637} a^{8} - \frac{6998942030393755989780206191386145561317309840316190}{18191459846216876035685364855544108881375167587159637} a^{7} + \frac{7645721110860190634897378362414527627388272052723147}{18191459846216876035685364855544108881375167587159637} a^{6} + \frac{1561128041083949400208580601548118986811090822080404}{18191459846216876035685364855544108881375167587159637} a^{5} - \frac{176633113185976635881564698028261332435783537794727}{18191459846216876035685364855544108881375167587159637} a^{4} + \frac{523017044745173101199985585833332614327786513267500}{1070085873306875060922668520914359345963245152185861} a^{3} - \frac{5900418841679595383309257779097671181775683695672114}{18191459846216876035685364855544108881375167587159637} a^{2} + \frac{1337236467064893097247158569332119529872130292881432}{18191459846216876035685364855544108881375167587159637} a - \frac{923328620013984941836483311643483402569742657733020}{2598779978030982290812194979363444125910738226737091}$

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:  $9$
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:  \( 580402652.6153839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.75449.1, 10.0.36252565459.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
19Data not computed