↑

Properties

Label 20.8.85263528582...3125.2
Degree $20$
Signature $[8, 6]$
Discriminant $3^{8}\cdot 5^{13}\cdot 239^{8}$
Root discriminant $39.49$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T144

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1045, -310, 12325, 14470, -9771, -7283, 15907, 4624, -12475, -5646, 2098, 696, -1054, -867, -178, 259, 162, -15, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 23*x^18 - 15*x^17 + 162*x^16 + 259*x^15 - 178*x^14 - 867*x^13 - 1054*x^12 + 696*x^11 + 2098*x^10 - 5646*x^9 - 12475*x^8 + 4624*x^7 + 15907*x^6 - 7283*x^5 - 9771*x^4 + 14470*x^3 + 12325*x^2 - 310*x - 1045)
 
gp: K = bnfinit(x^20 - 23*x^18 - 15*x^17 + 162*x^16 + 259*x^15 - 178*x^14 - 867*x^13 - 1054*x^12 + 696*x^11 + 2098*x^10 - 5646*x^9 - 12475*x^8 + 4624*x^7 + 15907*x^6 - 7283*x^5 - 9771*x^4 + 14470*x^3 + 12325*x^2 - 310*x - 1045, 1)
 

Normalized defining polynomial

\( x^{20} - 23 x^{18} - 15 x^{17} + 162 x^{16} + 259 x^{15} - 178 x^{14} - 867 x^{13} - 1054 x^{12} + 696 x^{11} + 2098 x^{10} - 5646 x^{9} - 12475 x^{8} + 4624 x^{7} + 15907 x^{6} - 7283 x^{5} - 9771 x^{4} + 14470 x^{3} + 12325 x^{2} - 310 x - 1045 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(85263528582144256599415283203125=3^{8}\cdot 5^{13}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.49$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{15} + \frac{2}{25} a^{14} - \frac{7}{25} a^{13} + \frac{2}{25} a^{12} - \frac{12}{25} a^{11} + \frac{7}{25} a^{10} - \frac{2}{25} a^{9} - \frac{4}{25} a^{8} + \frac{12}{25} a^{7} + \frac{1}{5} a^{6} - \frac{12}{25} a^{5} - \frac{11}{25} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{15} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{6}{25} a^{9} - \frac{7}{25} a^{8} + \frac{2}{25} a^{7} + \frac{8}{25} a^{6} + \frac{7}{25} a^{5} + \frac{4}{25} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{125} a^{18} + \frac{2}{125} a^{17} - \frac{2}{125} a^{15} + \frac{8}{125} a^{14} - \frac{13}{125} a^{13} + \frac{23}{125} a^{12} - \frac{18}{125} a^{11} + \frac{47}{125} a^{10} - \frac{2}{125} a^{9} + \frac{62}{125} a^{8} + \frac{3}{25} a^{7} + \frac{8}{125} a^{6} - \frac{2}{5} a^{5} + \frac{9}{125} a^{4} + \frac{7}{25} a^{3} + \frac{4}{25} a^{2} + \frac{2}{5} a + \frac{11}{25}$, $\frac{1}{7318482522245677991799280853702881625} a^{19} - \frac{3772769776029506781492679729257018}{7318482522245677991799280853702881625} a^{18} + \frac{9755909106116893651628327489269436}{1463696504449135598359856170740576325} a^{17} + \frac{43645874266997423210344943287619413}{7318482522245677991799280853702881625} a^{16} - \frac{531738124847337324133080670842728997}{7318482522245677991799280853702881625} a^{15} - \frac{91874042478905030633362515667794943}{7318482522245677991799280853702881625} a^{14} - \frac{585584610817929157458312582660605297}{7318482522245677991799280853702881625} a^{13} - \frac{3613724338553330711719096716032962973}{7318482522245677991799280853702881625} a^{12} - \frac{1914968683153757174752453885367194498}{7318482522245677991799280853702881625} a^{11} - \frac{3363364356836613970193254861690158187}{7318482522245677991799280853702881625} a^{10} - \frac{1212870237279268439710570441167972998}{7318482522245677991799280853702881625} a^{9} + \frac{1418610719070737556529280320270444}{58547860177965423934394246829623053} a^{8} + \frac{1126322917179209374704425328995023953}{7318482522245677991799280853702881625} a^{7} + \frac{51270500387863438904897359511250806}{292739300889827119671971234148115265} a^{6} - \frac{3214367300898741709481507989820241206}{7318482522245677991799280853702881625} a^{5} - \frac{696486551655161772709566299216608491}{1463696504449135598359856170740576325} a^{4} + \frac{520868317460139345207999150875676594}{1463696504449135598359856170740576325} a^{3} - \frac{120997272856332924503523035171203562}{292739300889827119671971234148115265} a^{2} + \frac{720431886334523221987485317058836176}{1463696504449135598359856170740576325} a - \frac{73372162340746577242686245149008992}{292739300889827119671971234148115265}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 200390600.563 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T144:

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 t20n144
Character table for t20n144 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed