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Properties

Label 20.0.16005732362...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{15}\cdot 11^{13}$
Root discriminant $45.73$
Ramified primes $5, 7, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33019779529, 988623515, -19912550031, -1732999488, 6066158054, 717065378, -1206080098, -153736813, 172316543, 21318004, -18368583, -2030017, 1467570, 132481, -85725, -5762, 3456, 155, -86, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529)
 
gp: K = bnfinit(x^20 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 86 x^{18} + 155 x^{17} + 3456 x^{16} - 5762 x^{15} - 85725 x^{14} + 132481 x^{13} + 1467570 x^{12} - 2030017 x^{11} - 18368583 x^{10} + 21318004 x^{9} + 172316543 x^{8} - 153736813 x^{7} - 1206080098 x^{6} + 717065378 x^{5} + 6066158054 x^{4} - 1732999488 x^{3} - 19912550031 x^{2} + 988623515 x + 33019779529 \)

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:  \(1600573236263365245152401689453125=5^{10}\cdot 7^{15}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.73$
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:  $5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{79} a^{18} - \frac{32}{79} a^{17} - \frac{17}{79} a^{16} + \frac{26}{79} a^{15} - \frac{31}{79} a^{14} - \frac{32}{79} a^{13} - \frac{27}{79} a^{12} + \frac{11}{79} a^{11} + \frac{14}{79} a^{10} + \frac{16}{79} a^{9} + \frac{25}{79} a^{8} + \frac{16}{79} a^{7} + \frac{2}{79} a^{6} + \frac{14}{79} a^{5} + \frac{35}{79} a^{4} - \frac{20}{79} a^{3} - \frac{27}{79} a^{2} + \frac{35}{79} a - \frac{8}{79}$, $\frac{1}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{19} - \frac{27479386370692438846852652017743915162684156840707667708010275085995539120535208}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{18} + \frac{2827987199818312910130744909849061263770209590406258195389937186890529874120398478}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{17} + \frac{6984121239980193737541813840344416736431690582102034846310794740418118553504438589}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{16} + \frac{5015426317861100902199842715202559434720221938917829303867874617665983887944641593}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{15} + \frac{14802382879901198702941469756587060617350872024654245908089245461767751548719339728}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{14} - \frac{11689225975606495890903028427690714215881891794046588207717736732035973875426300756}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{13} + \frac{12874434106311593029010504468744119568844176552112100244564072139024366649513544296}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{12} + \frac{9896626448570983636938942022458493602495547612622116588126492864676525676655650976}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{11} - \frac{8839549188225201461167993792165247091456857140745065635813293982085992959077961915}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{10} - \frac{4239449454408228556305459626211175029415225468096757787160487517931367437957228527}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{9} - \frac{6534523624063734893945601255400164055162241135529505686110702582802427733388288808}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{8} + \frac{1970976699160798586616270275870862752457330181780173284469262983944111244620603999}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{7} + \frac{6022707400217709919077622505241134105148752993452926750901326580327411449175684}{86888228819607603184945375760854759731462531002823044013328546841380992125556749} a^{6} + \frac{7957993696013798331384592535344060977355001237001417251128327337414571237010963687}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{5} - \frac{5908422911502527658931400626530733253576654376072464974388764927685221824264860371}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{4} - \frac{12192233389995617352403207554383222565251122980138997560273935534723950562027072547}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{3} + \frac{8588956528758704389277797998172161681511846824009764205604253076881874300577541801}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a^{2} - \frac{11528936456611334489323081925695338787767254743146086627438905664226923480870012454}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377} a + \frac{8602989276711898619098776960880896226707434178264865509262692549682531905822028366}{32409309349713635987984625158798825379835524064052995416971547971835110062832667377}$

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:  $9$
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:  \( 76205249.56986351 \) (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

\(\Q(\sqrt{-7}) \), 4.0.94325.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
5Data not computed
7Data not computed
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$