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Properties

Label 20.0.13610161416...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $113.94$
Ramified primes $2, 5, 7, 11$
Class number $5992888$ (GRH)
Class group $[11, 544808]$ (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![1432950026551, -75540326030, 686231001401, -39041186866, 152974017624, -9132285094, 20997115047, -1271790984, 1977453966, -117207804, 134567521, -7510492, 6764726, -339872, 250786, -10626, 6643, -210, 115, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 115*x^18 - 210*x^17 + 6643*x^16 - 10626*x^15 + 250786*x^14 - 339872*x^13 + 6764726*x^12 - 7510492*x^11 + 134567521*x^10 - 117207804*x^9 + 1977453966*x^8 - 1271790984*x^7 + 20997115047*x^6 - 9132285094*x^5 + 152974017624*x^4 - 39041186866*x^3 + 686231001401*x^2 - 75540326030*x + 1432950026551)
 
gp: K = bnfinit(x^20 - 2*x^19 + 115*x^18 - 210*x^17 + 6643*x^16 - 10626*x^15 + 250786*x^14 - 339872*x^13 + 6764726*x^12 - 7510492*x^11 + 134567521*x^10 - 117207804*x^9 + 1977453966*x^8 - 1271790984*x^7 + 20997115047*x^6 - 9132285094*x^5 + 152974017624*x^4 - 39041186866*x^3 + 686231001401*x^2 - 75540326030*x + 1432950026551, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 115 x^{18} - 210 x^{17} + 6643 x^{16} - 10626 x^{15} + 250786 x^{14} - 339872 x^{13} + 6764726 x^{12} - 7510492 x^{11} + 134567521 x^{10} - 117207804 x^{9} + 1977453966 x^{8} - 1271790984 x^{7} + 20997115047 x^{6} - 9132285094 x^{5} + 152974017624 x^{4} - 39041186866 x^{3} + 686231001401 x^{2} - 75540326030 x + 1432950026551 \)

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:  \(136101614161182402364761324912640000000000=2^{30}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.94$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(69,·)$, $\chi_{3080}(2689,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(1301,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(729,·)$, $\chi_{3080}(2589,·)$, $\chi_{3080}(741,·)$, $\chi_{3080}(1189,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(2869,·)$, $\chi_{3080}(2029,·)$, $\chi_{3080}(449,·)$, $\chi_{3080}(181,·)$, $\chi_{3080}(169,·)$, $\chi_{3080}(1849,·)$, $\chi_{3080}(1021,·)$, $\chi_{3080}(2421,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} - \frac{131}{331} a^{17} - \frac{152}{331} a^{16} + \frac{132}{331} a^{15} - \frac{165}{331} a^{14} - \frac{150}{331} a^{13} + \frac{63}{331} a^{12} - \frac{67}{331} a^{11} + \frac{35}{331} a^{10} - \frac{86}{331} a^{9} - \frac{72}{331} a^{8} - \frac{6}{331} a^{7} + \frac{165}{331} a^{6} + \frac{9}{331} a^{5} + \frac{104}{331} a^{4} - \frac{152}{331} a^{3} - \frac{121}{331} a^{2} + \frac{160}{331}$, $\frac{1}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{19} + \frac{80574746984535546216118910586218975315226011235224593041771874218863137295025848}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{18} - \frac{33976716460987125534931521754860595845857124713919292817217266074358828837366536570}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{17} - \frac{32158691188383156121737427833514322809026229373105527537895932611191047247029536219}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{16} + \frac{25005758631177081818059567532008755607355123986887501795203352035356996821972856651}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{15} + \frac{16878073020597200000559313306754555839112731823299967969188046226875547796745259609}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{14} - \frac{40541873535834735974351224172168940166816265990089501674230779601809997847615596215}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{13} + \frac{33042449113053047350426124520134207501587661086877279043598878803710576149505551099}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{12} + \frac{6224305168028778288695490938544607945067500026027161721888368652690048798563705742}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{11} - \frac{818893359522555789155751015811477465465284206818669746279224024882356771383165371}{2650669369624115487658134130621077792205347471581675958841531381955774577105319717} a^{10} + \frac{27745894436599240436504654934874368692475682317352050840589644846634514464246435682}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{9} + \frac{42612336872828599250903782347062271904221317004492251951203702596502641090632850016}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{8} + \frac{7843308213869995823402270052319307477198259408312748252632414333296496504075722751}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{7} + \frac{38159614593668879244731681821540092591946030068006349811994285076913805369750786628}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{6} - \frac{45457544439167904661189371103676015885439955778303552704777903344983767960548530534}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{5} - \frac{13465304699199168779463304070403190302484855949121589361735136985093809062639798593}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{4} - \frac{6014600466118807178921563266138996430203386348146310419997262687733574616476438860}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{3} - \frac{1975573167545847355954700921355026657685874181969286807673851455395343953516795482}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a^{2} - \frac{3335001926089704699143283900159288799487022694067913071351619163464776437150332103}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831} a + \frac{20149238378910849322432432485674386880201141046810577278659925841138258879313728757}{113978782893836965969299767616706345064829941278012066230185849424098306815528747831}$

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

Class group and class number

$C_{11}\times C_{544808}$, which has order $5992888$ (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:  \( 140644.5991815038 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.368919522607820800000.1, 10.0.118054247234502656.1

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 R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$