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Properties

Label 20.0.13936805290...4336.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $50.96$
Ramified primes $2, 7, 11$
Class number $1210$ (GRH)
Class group $[11, 110]$ (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![1908103, -1817414, 3509093, -2279050, 2267144, -960348, 672301, -233920, 160716, -64886, 31152, -3738, -1614, 746, 1124, -716, 6, 76, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 3*x^18 + 76*x^17 + 6*x^16 - 716*x^15 + 1124*x^14 + 746*x^13 - 1614*x^12 - 3738*x^11 + 31152*x^10 - 64886*x^9 + 160716*x^8 - 233920*x^7 + 672301*x^6 - 960348*x^5 + 2267144*x^4 - 2279050*x^3 + 3509093*x^2 - 1817414*x + 1908103)
 
gp: K = bnfinit(x^20 - 6*x^19 - 3*x^18 + 76*x^17 + 6*x^16 - 716*x^15 + 1124*x^14 + 746*x^13 - 1614*x^12 - 3738*x^11 + 31152*x^10 - 64886*x^9 + 160716*x^8 - 233920*x^7 + 672301*x^6 - 960348*x^5 + 2267144*x^4 - 2279050*x^3 + 3509093*x^2 - 1817414*x + 1908103, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 3 x^{18} + 76 x^{17} + 6 x^{16} - 716 x^{15} + 1124 x^{14} + 746 x^{13} - 1614 x^{12} - 3738 x^{11} + 31152 x^{10} - 64886 x^{9} + 160716 x^{8} - 233920 x^{7} + 672301 x^{6} - 960348 x^{5} + 2267144 x^{4} - 2279050 x^{3} + 3509093 x^{2} - 1817414 x + 1908103 \)

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:  \(13936805290105078002151559671054336=2^{30}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.96$
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:  \(616=2^{3}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{616}(1,·)$, $\chi_{616}(69,·)$, $\chi_{616}(181,·)$, $\chi_{616}(449,·)$, $\chi_{616}(265,·)$, $\chi_{616}(141,·)$, $\chi_{616}(405,·)$, $\chi_{616}(477,·)$, $\chi_{616}(97,·)$, $\chi_{616}(421,·)$, $\chi_{616}(433,·)$, $\chi_{616}(489,·)$, $\chi_{616}(225,·)$, $\chi_{616}(573,·)$, $\chi_{616}(113,·)$, $\chi_{616}(309,·)$, $\chi_{616}(169,·)$, $\chi_{616}(377,·)$, $\chi_{616}(125,·)$, $\chi_{616}(533,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{9}{23} a^{13} + \frac{7}{23} a^{12} - \frac{3}{23} a^{11} + \frac{6}{23} a^{10} - \frac{6}{23} a^{9} + \frac{1}{23} a^{8} + \frac{8}{23} a^{7} + \frac{4}{23} a^{6} + \frac{3}{23} a^{5} + \frac{10}{23} a^{4} + \frac{5}{23} a^{3} + \frac{4}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{345} a^{15} + \frac{1}{115} a^{14} - \frac{14}{69} a^{13} + \frac{54}{115} a^{12} - \frac{137}{345} a^{11} + \frac{9}{115} a^{10} - \frac{32}{345} a^{9} - \frac{7}{115} a^{8} - \frac{136}{345} a^{7} + \frac{16}{115} a^{6} - \frac{77}{345} a^{5} - \frac{49}{115} a^{4} - \frac{49}{345} a^{3} - \frac{36}{115} a^{2} - \frac{94}{345} a + \frac{1}{15}$, $\frac{1}{345} a^{16} - \frac{4}{345} a^{14} + \frac{4}{115} a^{13} - \frac{98}{345} a^{12} - \frac{44}{115} a^{11} - \frac{8}{345} a^{10} - \frac{2}{23} a^{9} + \frac{2}{345} a^{8} + \frac{7}{115} a^{7} + \frac{79}{345} a^{6} - \frac{12}{115} a^{5} + \frac{107}{345} a^{4} + \frac{1}{5} a^{3} - \frac{32}{69} a^{2} - \frac{26}{69} a - \frac{1}{5}$, $\frac{1}{345} a^{17} - \frac{2}{115} a^{14} + \frac{14}{115} a^{13} - \frac{13}{115} a^{12} - \frac{121}{345} a^{11} - \frac{34}{115} a^{10} + \frac{18}{115} a^{9} - \frac{31}{115} a^{8} - \frac{1}{23} a^{7} + \frac{12}{115} a^{6} + \frac{18}{115} a^{5} - \frac{43}{115} a^{4} - \frac{7}{15} a^{3} + \frac{8}{345} a^{2} + \frac{1}{69} a + \frac{4}{15}$, $\frac{1}{246593706910665} a^{18} + \frac{244976091812}{246593706910665} a^{17} + \frac{46181252951}{49318741382133} a^{16} - \frac{244648081006}{246593706910665} a^{15} + \frac{743051667889}{49318741382133} a^{14} - \frac{17500098533353}{49318741382133} a^{13} + \frac{1158697871169}{3573821839285} a^{12} - \frac{20899655460143}{82197902303555} a^{11} + \frac{6238424024132}{49318741382133} a^{10} - \frac{9649254983885}{49318741382133} a^{9} - \frac{113417916381736}{246593706910665} a^{8} - \frac{109186239721}{2770715807985} a^{7} + \frac{79269819523321}{246593706910665} a^{6} - \frac{3146045049781}{246593706910665} a^{5} + \frac{31255055911822}{82197902303555} a^{4} - \frac{73505343097489}{246593706910665} a^{3} - \frac{258582071951}{2770715807985} a^{2} - \frac{21498026686621}{82197902303555} a + \frac{5057750412613}{10721465517855}$, $\frac{1}{218720041090197910540618611240999795705} a^{19} + \frac{347679397586777541995071}{218720041090197910540618611240999795705} a^{18} - \frac{48125193406742859648920648425344788}{218720041090197910540618611240999795705} a^{17} + \frac{20474625270216008338204531393874660}{43744008218039582108123722248199959141} a^{16} - \frac{3788348823304184948015868219709058}{72906680363399303513539537080333265235} a^{15} + \frac{3629000135747109745155780313392925922}{218720041090197910540618611240999795705} a^{14} + \frac{57786428012280534651485267341784650766}{218720041090197910540618611240999795705} a^{13} - \frac{30972815444113575729256681007791610538}{72906680363399303513539537080333265235} a^{12} - \frac{65558192598924741983056342873423845352}{218720041090197910540618611240999795705} a^{11} - \frac{84860426711675614741219835004794698181}{218720041090197910540618611240999795705} a^{10} - \frac{4845699749430745073630037283426313861}{14581336072679860702707907416066653047} a^{9} - \frac{47315423402729532536355280548879900118}{218720041090197910540618611240999795705} a^{8} - \frac{22810008503840292849565911577036379443}{72906680363399303513539537080333265235} a^{7} + \frac{1598113137331231809128820403493841001}{218720041090197910540618611240999795705} a^{6} - \frac{31844942894412304890094634325734354618}{218720041090197910540618611240999795705} a^{5} - \frac{107608811497013050405973052044071830454}{218720041090197910540618611240999795705} a^{4} + \frac{19177790385011177664456461516398529503}{43744008218039582108123722248199959141} a^{3} + \frac{50691730576916092328469324496390484153}{218720041090197910540618611240999795705} a^{2} + \frac{872952193372659227757559777791006423}{3169855667973882761458240742623185445} a + \frac{22669573365256783998271861877608027}{82691886990623028559780193285822229}$

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

Class group and class number

$C_{11}\times C_{110}$, which has order $1210$ (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:  \( 530208.250733 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1, 10.0.3602729712967.1, 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{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$11.10.8.5$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}$