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Properties

Label 20.0.16468295313...2944.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $45.80$
Ramified primes $2, 7, 11$
Class number $620$ (GRH)
Class group $[2, 310]$ (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![558097, -328784, 532837, -466344, 404164, -279504, 228417, -144344, 84390, -52602, 32440, -15070, 6858, -3886, 1932, -644, 208, -102, 43, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 43*x^18 - 102*x^17 + 208*x^16 - 644*x^15 + 1932*x^14 - 3886*x^13 + 6858*x^12 - 15070*x^11 + 32440*x^10 - 52602*x^9 + 84390*x^8 - 144344*x^7 + 228417*x^6 - 279504*x^5 + 404164*x^4 - 466344*x^3 + 532837*x^2 - 328784*x + 558097)
 
gp: K = bnfinit(x^20 - 10*x^19 + 43*x^18 - 102*x^17 + 208*x^16 - 644*x^15 + 1932*x^14 - 3886*x^13 + 6858*x^12 - 15070*x^11 + 32440*x^10 - 52602*x^9 + 84390*x^8 - 144344*x^7 + 228417*x^6 - 279504*x^5 + 404164*x^4 - 466344*x^3 + 532837*x^2 - 328784*x + 558097, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 43 x^{18} - 102 x^{17} + 208 x^{16} - 644 x^{15} + 1932 x^{14} - 3886 x^{13} + 6858 x^{12} - 15070 x^{11} + 32440 x^{10} - 52602 x^{9} + 84390 x^{8} - 144344 x^{7} + 228417 x^{6} - 279504 x^{5} + 404164 x^{4} - 466344 x^{3} + 532837 x^{2} - 328784 x + 558097 \)

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:  \(1646829531350307068613612031442944=2^{20}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.80$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(195,·)$, $\chi_{308}(69,·)$, $\chi_{308}(211,·)$, $\chi_{308}(225,·)$, $\chi_{308}(265,·)$, $\chi_{308}(139,·)$, $\chi_{308}(141,·)$, $\chi_{308}(83,·)$, $\chi_{308}(97,·)$, $\chi_{308}(167,·)$, $\chi_{308}(169,·)$, $\chi_{308}(43,·)$, $\chi_{308}(239,·)$, $\chi_{308}(113,·)$, $\chi_{308}(307,·)$, $\chi_{308}(181,·)$, $\chi_{308}(183,·)$, $\chi_{308}(125,·)$, $\chi_{308}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{325898554925798453281} a^{18} - \frac{9}{325898554925798453281} a^{17} + \frac{124926428138162596695}{325898554925798453281} a^{16} - \frac{21715760327905413513}{325898554925798453281} a^{15} - \frac{10200487382433529405}{325898554925798453281} a^{14} - \frac{37418614973318238323}{325898554925798453281} a^{13} + \frac{116644691487958662254}{325898554925798453281} a^{12} - \frac{60848306954016601031}{325898554925798453281} a^{11} - \frac{26136976627248001667}{325898554925798453281} a^{10} + \frac{48839168791147025580}{325898554925798453281} a^{9} - \frac{29837503090115034957}{325898554925798453281} a^{8} + \frac{8388794001510463429}{325898554925798453281} a^{7} - \frac{147926378092951520771}{325898554925798453281} a^{6} + \frac{93961219465098694537}{325898554925798453281} a^{5} - \frac{141157135312716032951}{325898554925798453281} a^{4} + \frac{22667820723189935301}{325898554925798453281} a^{3} - \frac{113260375517194903568}{325898554925798453281} a^{2} - \frac{152825139254966554883}{325898554925798453281} a + \frac{125824915343831936928}{325898554925798453281}$, $\frac{1}{233088838555474643957183539} a^{19} + \frac{357600}{233088838555474643957183539} a^{18} + \frac{31270743068668351348596726}{233088838555474643957183539} a^{17} + \frac{11962399975067914589555086}{233088838555474643957183539} a^{16} + \frac{10010049244283441381259042}{233088838555474643957183539} a^{15} + \frac{109377706119966367844059204}{233088838555474643957183539} a^{14} - \frac{5151200629760685589082360}{233088838555474643957183539} a^{13} + \frac{96558598438473076936049145}{233088838555474643957183539} a^{12} + \frac{8605343618160795159907348}{233088838555474643957183539} a^{11} + \frac{83956357546956728700975272}{233088838555474643957183539} a^{10} + \frac{108854095145577534623114564}{233088838555474643957183539} a^{9} - \frac{1668302848273373493052748}{5420670664080805673422873} a^{8} + \frac{649379929583203919797918}{233088838555474643957183539} a^{7} + \frac{109604063956092824862162871}{233088838555474643957183539} a^{6} - \frac{45179850600428722765233453}{233088838555474643957183539} a^{5} - \frac{4246092636785798659836355}{233088838555474643957183539} a^{4} - \frac{55435967306033841050311234}{233088838555474643957183539} a^{3} - \frac{42499862899498174359856482}{233088838555474643957183539} a^{2} - \frac{73751781906005186643110667}{233088838555474643957183539} a - \frac{2126216144898402744179405}{5420670664080805673422873}$

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

Class group and class number

$C_{2}\times C_{310}$, which has order $620$ (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

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-7}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 10.0.40581147486860288.1, 10.0.3602729712967.1, \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$