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Properties

Label 20.0.51609443851...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{32}\cdot 17^{10}$
Root discriminant $216.59$
Ramified primes $2, 5, 17$
Class number $320762200$ (GRH)
Class group $[10, 32076220]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![74892112393, 13607859580, 39412521260, 5730189260, 9636856145, 1106055084, 1463431730, 125875400, 155212695, 9163340, 12271392, 434320, 750950, 11480, 36230, -44, 1365, -20, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 40*x^18 - 20*x^17 + 1365*x^16 - 44*x^15 + 36230*x^14 + 11480*x^13 + 750950*x^12 + 434320*x^11 + 12271392*x^10 + 9163340*x^9 + 155212695*x^8 + 125875400*x^7 + 1463431730*x^6 + 1106055084*x^5 + 9636856145*x^4 + 5730189260*x^3 + 39412521260*x^2 + 13607859580*x + 74892112393)
 
gp: K = bnfinit(x^20 + 40*x^18 - 20*x^17 + 1365*x^16 - 44*x^15 + 36230*x^14 + 11480*x^13 + 750950*x^12 + 434320*x^11 + 12271392*x^10 + 9163340*x^9 + 155212695*x^8 + 125875400*x^7 + 1463431730*x^6 + 1106055084*x^5 + 9636856145*x^4 + 5730189260*x^3 + 39412521260*x^2 + 13607859580*x + 74892112393, 1)
 

Normalized defining polynomial

\( x^{20} + 40 x^{18} - 20 x^{17} + 1365 x^{16} - 44 x^{15} + 36230 x^{14} + 11480 x^{13} + 750950 x^{12} + 434320 x^{11} + 12271392 x^{10} + 9163340 x^{9} + 155212695 x^{8} + 125875400 x^{7} + 1463431730 x^{6} + 1106055084 x^{5} + 9636856145 x^{4} + 5730189260 x^{3} + 39412521260 x^{2} + 13607859580 x + 74892112393 \)

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:  \(51609443851494400000000000000000000000000000000=2^{40}\cdot 5^{32}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.59$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3400=2^{3}\cdot 5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3400}(1,·)$, $\chi_{3400}(3331,·)$, $\chi_{3400}(2311,·)$, $\chi_{3400}(1291,·)$, $\chi_{3400}(2381,·)$, $\chi_{3400}(271,·)$, $\chi_{3400}(1361,·)$, $\chi_{3400}(341,·)$, $\chi_{3400}(2651,·)$, $\chi_{3400}(1631,·)$, $\chi_{3400}(2721,·)$, $\chi_{3400}(611,·)$, $\chi_{3400}(1701,·)$, $\chi_{3400}(681,·)$, $\chi_{3400}(2991,·)$, $\chi_{3400}(1971,·)$, $\chi_{3400}(3061,·)$, $\chi_{3400}(951,·)$, $\chi_{3400}(2041,·)$, $\chi_{3400}(1021,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{441} a^{15} + \frac{2}{63} a^{13} + \frac{1}{21} a^{12} + \frac{4}{63} a^{11} + \frac{1}{21} a^{10} - \frac{2}{441} a^{9} - \frac{1}{21} a^{8} + \frac{2}{63} a^{7} + \frac{4}{63} a^{6} - \frac{11}{63} a^{5} - \frac{31}{63} a^{4} + \frac{1}{441} a^{3} + \frac{31}{63} a^{2} + \frac{10}{63} a - \frac{1}{9}$, $\frac{1}{441} a^{16} - \frac{4}{441} a^{14} + \frac{1}{21} a^{13} + \frac{4}{63} a^{12} + \frac{1}{21} a^{11} - \frac{2}{441} a^{10} - \frac{1}{21} a^{9} - \frac{13}{441} a^{8} + \frac{4}{63} a^{7} - \frac{11}{63} a^{6} - \frac{31}{63} a^{5} + \frac{1}{441} a^{4} + \frac{31}{63} a^{3} + \frac{115}{441} a^{2} - \frac{1}{9} a$, $\frac{1}{441} a^{17} + \frac{1}{147} a^{14} + \frac{1}{21} a^{13} - \frac{1}{21} a^{12} - \frac{16}{441} a^{11} - \frac{1}{21} a^{9} - \frac{20}{441} a^{8} - \frac{1}{21} a^{7} + \frac{1}{21} a^{6} - \frac{181}{441} a^{5} - \frac{1}{3} a^{4} + \frac{17}{63} a^{3} - \frac{9}{49} a^{2} - \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{122325197548695674330763} a^{18} - \frac{18757092404211744022}{40775065849565224776921} a^{17} + \frac{13462710001486160573}{17475028221242239190109} a^{16} - \frac{95980848915602112314}{122325197548695674330763} a^{15} - \frac{797350347533807597429}{122325197548695674330763} a^{14} + \frac{522073024714639980887}{17475028221242239190109} a^{13} + \frac{3012645260355365848519}{122325197548695674330763} a^{12} - \frac{3535095919860439597718}{122325197548695674330763} a^{11} - \frac{597772127262791048836}{17475028221242239190109} a^{10} + \frac{5425351862417196118631}{122325197548695674330763} a^{9} + \frac{5755967846099739293236}{122325197548695674330763} a^{8} - \frac{754885936007982656705}{17475028221242239190109} a^{7} - \frac{16693630659603460323529}{122325197548695674330763} a^{6} + \frac{56620101249697631603660}{122325197548695674330763} a^{5} - \frac{1733892815875381169242}{5825009407080746396703} a^{4} + \frac{3602503465325808697637}{13591688616521741592307} a^{3} - \frac{52642803787937880024617}{122325197548695674330763} a^{2} - \frac{1664838071746103675864}{17475028221242239190109} a + \frac{416583486407901022058}{2496432603034605598587}$, $\frac{1}{2683837204052576070971863824626046635259615411309} a^{19} - \frac{1640324865717669440370907}{894612401350858690323954608208682211753205137103} a^{18} + \frac{86468690758008186919594149501541988085146765}{127801771621551241474850658315526030250457876729} a^{17} - \frac{1100079736882469086294691084121388730110495154}{2683837204052576070971863824626046635259615411309} a^{16} + \frac{2832390360575300114930254311110711261233106189}{2683837204052576070971863824626046635259615411309} a^{15} + \frac{2052471203107909942250976828928055863417049171}{383405314864653724424551974946578090751373630187} a^{14} + \frac{16245104210956489447399796004255324719567554318}{894612401350858690323954608208682211753205137103} a^{13} + \frac{159504227797846545506664277251795342470218222084}{2683837204052576070971863824626046635259615411309} a^{12} - \frac{50774809648860087044455960901132974111901140}{54772187837807674917793139278082584393053375741} a^{11} - \frac{74281633439680705021917280338233841082924979317}{2683837204052576070971863824626046635259615411309} a^{10} + \frac{165446414320254133534949383209692600587647453809}{2683837204052576070971863824626046635259615411309} a^{9} - \frac{518011655200119023748954891429596575190538979}{54772187837807674917793139278082584393053375741} a^{8} - \frac{14072813092563375733043644543450087088523030500}{2683837204052576070971863824626046635259615411309} a^{7} + \frac{30953942844512481739102275097827816207204587602}{894612401350858690323954608208682211753205137103} a^{6} - \frac{59550412587283747982229491523506920336908326969}{127801771621551241474850658315526030250457876729} a^{5} - \frac{1251999853439432191672867961356731577615270742275}{2683837204052576070971863824626046635259615411309} a^{4} + \frac{15394768066728771894794812186216304797715824315}{2683837204052576070971863824626046635259615411309} a^{3} + \frac{143812589085781681058874361123362140905801719056}{383405314864653724424551974946578090751373630187} a^{2} - \frac{13241646493488641069586074872276901914188972449}{54772187837807674917793139278082584393053375741} a - \frac{2715927175242042922457237238533208860885354114}{7824598262543953559684734182583226341864767963}$

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

Class group and class number

$C_{10}\times C_{32076220}$, which has order $320762200$ (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:  \( 42294001.73672045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{2}, \sqrt{-17})\), 5.5.390625.1, 10.10.5000000000000000.1, 10.0.7099285000000000000000.1, 10.0.221852656250000000000.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
$17$17.10.5.1$x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
17.10.5.1$x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$