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Properties

Label 20.0.33618693882...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number $15928$ (GRH)
Class group $[2, 7964]$ (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![51338101, -38299500, 55028485, -38415016, 32733776, -19605652, 12787473, -6710840, 3590458, -1649418, 755996, -304054, 119190, -41798, 14148, -4180, 1160, -282, 63, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101)
 
gp: K = bnfinit(x^20 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 63 x^{18} - 282 x^{17} + 1160 x^{16} - 4180 x^{15} + 14148 x^{14} - 41798 x^{13} + 119190 x^{12} - 304054 x^{11} + 755996 x^{10} - 1649418 x^{9} + 3590458 x^{8} - 6710840 x^{7} + 12787473 x^{6} - 19605652 x^{5} + 32733776 x^{4} - 38415016 x^{3} + 55028485 x^{2} - 38299500 x + 51338101 \)

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:  \(3361869388230684433628866560000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(389,·)$, $\chi_{660}(391,·)$, $\chi_{660}(269,·)$, $\chi_{660}(271,·)$, $\chi_{660}(659,·)$, $\chi_{660}(151,·)$, $\chi_{660}(89,·)$, $\chi_{660}(479,·)$, $\chi_{660}(421,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(239,·)$, $\chi_{660}(449,·)$, $\chi_{660}(211,·)$, $\chi_{660}(181,·)$, $\chi_{660}(571,·)$, $\chi_{660}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{34605839688798883943040241} a^{18} - \frac{9}{34605839688798883943040241} a^{17} - \frac{2209264105144059104216443}{34605839688798883943040241} a^{16} - \frac{16931726847646411109308493}{34605839688798883943040241} a^{15} + \frac{9662586070864218463136743}{34605839688798883943040241} a^{14} + \frac{3729159360569919541179146}{34605839688798883943040241} a^{13} - \frac{16487300888405307592118578}{34605839688798883943040241} a^{12} - \frac{10529124304511693123826269}{34605839688798883943040241} a^{11} - \frac{13411070524892239379515955}{34605839688798883943040241} a^{10} + \frac{9233213185738372511382598}{34605839688798883943040241} a^{9} + \frac{2604026821261913987692273}{34605839688798883943040241} a^{8} + \frac{11345759218696179300564748}{34605839688798883943040241} a^{7} + \frac{11963458113833524890251086}{34605839688798883943040241} a^{6} + \frac{3998458283652468698366957}{34605839688798883943040241} a^{5} + \frac{14622088029749767298450235}{34605839688798883943040241} a^{4} - \frac{7325658010027551421763887}{34605839688798883943040241} a^{3} - \frac{3223528667576436714407276}{34605839688798883943040241} a^{2} + \frac{2958924263837333754133123}{34605839688798883943040241} a + \frac{14373171098349861010416340}{34605839688798883943040241}$, $\frac{1}{231656301088536472159579455847499} a^{19} + \frac{3347060}{231656301088536472159579455847499} a^{18} + \frac{22915396533383806221995081566039}{231656301088536472159579455847499} a^{17} - \frac{86347280493266590141088899201945}{231656301088536472159579455847499} a^{16} - \frac{73014325390353605260872245113006}{231656301088536472159579455847499} a^{15} - \frac{61370644945372189940729821468249}{231656301088536472159579455847499} a^{14} + \frac{33913341358833793242695830397242}{231656301088536472159579455847499} a^{13} + \frac{46988109403693917501078900247986}{231656301088536472159579455847499} a^{12} - \frac{45168231053381688949152823801161}{231656301088536472159579455847499} a^{11} + \frac{32868576501050498486607743462977}{231656301088536472159579455847499} a^{10} - \frac{59198743722605173726207502572237}{231656301088536472159579455847499} a^{9} + \frac{60388547289214851813351430038483}{231656301088536472159579455847499} a^{8} - \frac{96496736206990652823586244185994}{231656301088536472159579455847499} a^{7} - \frac{24005895100280650830654481428427}{231656301088536472159579455847499} a^{6} - \frac{74433098279117587775808484235465}{231656301088536472159579455847499} a^{5} - \frac{5498631202369184834081330798930}{231656301088536472159579455847499} a^{4} - \frac{108273196779278720183869454937258}{231656301088536472159579455847499} a^{3} - \frac{14151569481576765787174653435758}{231656301088536472159579455847499} a^{2} - \frac{69682114555336726912435420204394}{231656301088536472159579455847499} a + \frac{76734949574908182907590444229649}{231656301088536472159579455847499}$

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

Class group and class number

$C_{2}\times C_{7964}$, which has order $15928$ (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:  \( 281202.490766 \) (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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{11}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 10.0.1833540124521600000.1, \(\Q(\zeta_{44})^+\), 10.0.162778775259375.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
3Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed