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Properties

Label 20.8.13156433165...8125.2
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 401^{11}$
Root discriminant $90.36$
Ramified primes $5, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2100751929, 2038938147, 215667329, -585134531, 169584681, -132536173, 98159604, -27484603, 1936911, 563814, -51244, 154704, -143330, 23621, -64, 1781, -208, -119, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 25*x^18 - 119*x^17 - 208*x^16 + 1781*x^15 - 64*x^14 + 23621*x^13 - 143330*x^12 + 154704*x^11 - 51244*x^10 + 563814*x^9 + 1936911*x^8 - 27484603*x^7 + 98159604*x^6 - 132536173*x^5 + 169584681*x^4 - 585134531*x^3 + 215667329*x^2 + 2038938147*x - 2100751929)
 
gp: K = bnfinit(x^20 - 6*x^19 + 25*x^18 - 119*x^17 - 208*x^16 + 1781*x^15 - 64*x^14 + 23621*x^13 - 143330*x^12 + 154704*x^11 - 51244*x^10 + 563814*x^9 + 1936911*x^8 - 27484603*x^7 + 98159604*x^6 - 132536173*x^5 + 169584681*x^4 - 585134531*x^3 + 215667329*x^2 + 2038938147*x - 2100751929, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 25 x^{18} - 119 x^{17} - 208 x^{16} + 1781 x^{15} - 64 x^{14} + 23621 x^{13} - 143330 x^{12} + 154704 x^{11} - 51244 x^{10} + 563814 x^{9} + 1936911 x^{8} - 27484603 x^{7} + 98159604 x^{6} - 132536173 x^{5} + 169584681 x^{4} - 585134531 x^{3} + 215667329 x^{2} + 2038938147 x - 2100751929 \)

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:  \(1315643316557894633135034313995361328125=5^{15}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.36$
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:  $5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{4}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{2}{9} a$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{2}{27} a^{9} + \frac{1}{27} a^{8} - \frac{5}{27} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{5}{27} a^{4} + \frac{7}{27} a^{3} - \frac{4}{27} a^{2} - \frac{4}{9} a$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{27} a^{10} - \frac{2}{27} a^{9} + \frac{1}{27} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{13}{27} a^{5} + \frac{13}{27} a^{4} - \frac{13}{27} a^{3} + \frac{2}{9} a$, $\frac{1}{243} a^{18} - \frac{1}{81} a^{17} - \frac{1}{81} a^{16} + \frac{10}{243} a^{15} - \frac{13}{243} a^{14} + \frac{13}{243} a^{13} + \frac{11}{81} a^{12} + \frac{10}{243} a^{11} + \frac{10}{243} a^{10} - \frac{4}{243} a^{9} - \frac{11}{243} a^{8} - \frac{38}{243} a^{7} + \frac{80}{243} a^{6} + \frac{10}{243} a^{5} - \frac{116}{243} a^{4} - \frac{56}{243} a^{3} - \frac{13}{243} a^{2} - \frac{1}{27} a + \frac{1}{3}$, $\frac{1}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{19} - \frac{481933389399127202903111121150111066503461944363018172454328170672435681}{863577364388510322348592425210898089832876186215145723418352861999490857447} a^{18} + \frac{10894581352077356618621117011458253583465226941948903258966555356524344543}{863577364388510322348592425210898089832876186215145723418352861999490857447} a^{17} + \frac{24660162464822063838261715975022213643888409969390982634675701575546892491}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{16} + \frac{102395801930875042424617309906856056740149004310707631392175433037104576753}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{15} + \frac{132681144121736627117332926068142218111189748437551102090715636340005709319}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{14} + \frac{42123219555603997358401140375136033674067200647022507258988591348951337866}{863577364388510322348592425210898089832876186215145723418352861999490857447} a^{13} - \frac{334277119476851507583084258062439299207578273560641471584027346410636639742}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{12} + \frac{7047032983620117479600427955083325568822170740309936701109996714245206192}{70019786301771107217453439881964709986449420503930734331217799621580339793} a^{11} + \frac{229587129892904021668986157474039811879941456996410385726519927750777001810}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{10} + \frac{351054500324072870467483864435148816152314633545866676030065943633040068083}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{9} - \frac{416426915086206566269967540080280241231741108849519950513183289018330887312}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{8} + \frac{1157179734059291563011290175636093922372422034449914586345621042704542550114}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{7} + \frac{136487550926825984283240487638346784768124671316008297299219681581050995315}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{6} - \frac{1139121212975518787896019504577979163651956564672120217526553320258311900762}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{5} + \frac{36767293930410016171689317317180529905006361513783566429064248633015776193}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{4} - \frac{479180412589741007707858842656909353501397793395643231681051496865238519145}{2590732093165530967045777275632694269498628558645437170255058585998472572341} a^{3} + \frac{192210603399614556720813585860293390738436541859854640099606650604089796732}{863577364388510322348592425210898089832876186215145723418352861999490857447} a^{2} - \frac{34827920702190999428654132016366184591217637623833268992566710905901904173}{95953040487612258038732491690099787759208465135016191490928095777721206383} a + \frac{3705079247619173129853716202098046258145217681649061642743071004458099414}{10661448943068028670970276854455531973245385015001799054547566197524578487}$

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:  \( 1139732818500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 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 $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
401Data not computed