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Properties

Label 20.0.12015805468...2769.2
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 43^{10}$
Root discriminant $56.75$
Ramified primes $11, 43$
Class number $1200$ (GRH)
Class group $[2, 2, 10, 30]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25937424601, -2357947691, -2143588810, 409230591, 157668929, -51536320, -9648419, 5562249, 371470, -539429, 15269, -49039, 3070, 4179, -659, -320, 89, 21, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 10*x^18 + 21*x^17 + 89*x^16 - 320*x^15 - 659*x^14 + 4179*x^13 + 3070*x^12 - 49039*x^11 + 15269*x^10 - 539429*x^9 + 371470*x^8 + 5562249*x^7 - 9648419*x^6 - 51536320*x^5 + 157668929*x^4 + 409230591*x^3 - 2143588810*x^2 - 2357947691*x + 25937424601)
 
gp: K = bnfinit(x^20 - x^19 - 10*x^18 + 21*x^17 + 89*x^16 - 320*x^15 - 659*x^14 + 4179*x^13 + 3070*x^12 - 49039*x^11 + 15269*x^10 - 539429*x^9 + 371470*x^8 + 5562249*x^7 - 9648419*x^6 - 51536320*x^5 + 157668929*x^4 + 409230591*x^3 - 2143588810*x^2 - 2357947691*x + 25937424601, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 10 x^{18} + 21 x^{17} + 89 x^{16} - 320 x^{15} - 659 x^{14} + 4179 x^{13} + 3070 x^{12} - 49039 x^{11} + 15269 x^{10} - 539429 x^{9} + 371470 x^{8} + 5562249 x^{7} - 9648419 x^{6} - 51536320 x^{5} + 157668929 x^{4} + 409230591 x^{3} - 2143588810 x^{2} - 2357947691 x + 25937424601 \)

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:  \(120158054683860237850976148859242769=11^{18}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.75$
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:  $11, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(473=11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{473}(128,·)$, $\chi_{473}(1,·)$, $\chi_{473}(130,·)$, $\chi_{473}(259,·)$, $\chi_{473}(388,·)$, $\chi_{473}(257,·)$, $\chi_{473}(343,·)$, $\chi_{473}(386,·)$, $\chi_{473}(216,·)$, $\chi_{473}(85,·)$, $\chi_{473}(214,·)$, $\chi_{473}(87,·)$, $\chi_{473}(472,·)$, $\chi_{473}(345,·)$, $\chi_{473}(42,·)$, $\chi_{473}(171,·)$, $\chi_{473}(300,·)$, $\chi_{473}(173,·)$, $\chi_{473}(302,·)$, $\chi_{473}(431,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{167959} a^{11} - \frac{1}{11} a^{10} + \frac{1}{11} a^{9} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} - \frac{1}{11} a^{6} + \frac{1}{11} a^{5} - \frac{1}{11} a^{4} + \frac{1}{11} a^{3} - \frac{1}{11} a^{2} + \frac{1}{11} a - \frac{3232}{15269}$, $\frac{1}{1847549} a^{12} - \frac{1}{1847549} a^{11} - \frac{32}{121} a^{10} + \frac{43}{121} a^{9} - \frac{54}{121} a^{8} - \frac{56}{121} a^{7} + \frac{45}{121} a^{6} - \frac{34}{121} a^{5} + \frac{23}{121} a^{4} - \frac{12}{121} a^{3} + \frac{1}{121} a^{2} - \frac{49039}{167959} a + \frac{3070}{15269}$, $\frac{1}{20323039} a^{13} - \frac{1}{20323039} a^{12} - \frac{10}{20323039} a^{11} - \frac{320}{1331} a^{10} - \frac{659}{1331} a^{9} + \frac{186}{1331} a^{8} + \frac{408}{1331} a^{7} + \frac{208}{1331} a^{6} + \frac{628}{1331} a^{5} - \frac{254}{1331} a^{4} + \frac{1}{1331} a^{3} - \frac{49039}{1847549} a^{2} + \frac{3070}{167959} a + \frac{4179}{15269}$, $\frac{1}{223553429} a^{14} - \frac{1}{223553429} a^{13} - \frac{10}{223553429} a^{12} + \frac{21}{223553429} a^{11} + \frac{3334}{14641} a^{10} + \frac{186}{14641} a^{9} + \frac{7063}{14641} a^{8} + \frac{5532}{14641} a^{7} + \frac{4621}{14641} a^{6} - \frac{6909}{14641} a^{5} + \frac{1}{14641} a^{4} - \frac{49039}{20323039} a^{3} + \frac{3070}{1847549} a^{2} + \frac{4179}{167959} a - \frac{659}{15269}$, $\frac{1}{2459087719} a^{15} - \frac{1}{2459087719} a^{14} - \frac{10}{2459087719} a^{13} + \frac{21}{2459087719} a^{12} + \frac{89}{2459087719} a^{11} + \frac{58750}{161051} a^{10} + \frac{65627}{161051} a^{9} - \frac{67673}{161051} a^{8} - \frac{10020}{161051} a^{7} - \frac{50832}{161051} a^{6} + \frac{1}{161051} a^{5} - \frac{49039}{223553429} a^{4} + \frac{3070}{20323039} a^{3} + \frac{4179}{1847549} a^{2} - \frac{659}{167959} a - \frac{320}{15269}$, $\frac{1}{27049964909} a^{16} - \frac{1}{27049964909} a^{15} - \frac{10}{27049964909} a^{14} + \frac{21}{27049964909} a^{13} + \frac{89}{27049964909} a^{12} - \frac{320}{27049964909} a^{11} + \frac{709831}{1771561} a^{10} + \frac{415480}{1771561} a^{9} + \frac{634184}{1771561} a^{8} + \frac{110219}{1771561} a^{7} + \frac{1}{1771561} a^{6} - \frac{49039}{2459087719} a^{5} + \frac{3070}{223553429} a^{4} + \frac{4179}{20323039} a^{3} - \frac{659}{1847549} a^{2} - \frac{320}{167959} a + \frac{89}{15269}$, $\frac{1}{297549613999} a^{17} - \frac{1}{297549613999} a^{16} - \frac{10}{297549613999} a^{15} + \frac{21}{297549613999} a^{14} + \frac{89}{297549613999} a^{13} - \frac{320}{297549613999} a^{12} - \frac{659}{297549613999} a^{11} + \frac{9273285}{19487171} a^{10} + \frac{2405745}{19487171} a^{9} - \frac{6976025}{19487171} a^{8} + \frac{1}{19487171} a^{7} - \frac{49039}{27049964909} a^{6} + \frac{3070}{2459087719} a^{5} + \frac{4179}{223553429} a^{4} - \frac{659}{20323039} a^{3} - \frac{320}{1847549} a^{2} + \frac{89}{167959} a + \frac{21}{15269}$, $\frac{1}{3273045753989} a^{18} - \frac{1}{3273045753989} a^{17} - \frac{10}{3273045753989} a^{16} + \frac{21}{3273045753989} a^{15} + \frac{89}{3273045753989} a^{14} - \frac{320}{3273045753989} a^{13} - \frac{659}{3273045753989} a^{12} + \frac{4179}{3273045753989} a^{11} - \frac{75542939}{214358881} a^{10} - \frac{26463196}{214358881} a^{9} + \frac{1}{214358881} a^{8} - \frac{49039}{297549613999} a^{7} + \frac{3070}{27049964909} a^{6} + \frac{4179}{2459087719} a^{5} - \frac{659}{223553429} a^{4} - \frac{320}{20323039} a^{3} + \frac{89}{1847549} a^{2} + \frac{21}{167959} a - \frac{10}{15269}$, $\frac{1}{36003503293879} a^{19} - \frac{1}{36003503293879} a^{18} - \frac{10}{36003503293879} a^{17} + \frac{21}{36003503293879} a^{16} + \frac{89}{36003503293879} a^{15} - \frac{320}{36003503293879} a^{14} - \frac{659}{36003503293879} a^{13} + \frac{4179}{36003503293879} a^{12} + \frac{3070}{36003503293879} a^{11} + \frac{830972328}{2357947691} a^{10} + \frac{1}{2357947691} a^{9} - \frac{49039}{3273045753989} a^{8} + \frac{3070}{297549613999} a^{7} + \frac{4179}{27049964909} a^{6} - \frac{659}{2459087719} a^{5} - \frac{320}{223553429} a^{4} + \frac{89}{20323039} a^{3} + \frac{21}{1847549} a^{2} - \frac{10}{167959} a - \frac{1}{15269}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{30}$, which has order $1200$ (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:  \( -\frac{49039}{36003503293879} a^{19} + \frac{49039}{36003503293879} a^{18} + \frac{490390}{36003503293879} a^{17} - \frac{1029819}{36003503293879} a^{16} - \frac{4364471}{36003503293879} a^{15} + \frac{15692480}{36003503293879} a^{14} + \frac{32316701}{36003503293879} a^{13} - \frac{204933981}{36003503293879} a^{12} - \frac{150549730}{36003503293879} a^{11} + \frac{3070}{2357947691} a^{10} - \frac{49039}{2357947691} a^{9} + \frac{2404823521}{3273045753989} a^{8} - \frac{150549730}{297549613999} a^{7} - \frac{204933981}{27049964909} a^{6} + \frac{32316701}{2459087719} a^{5} + \frac{15692480}{223553429} a^{4} - \frac{4364471}{20323039} a^{3} - \frac{1029819}{1847549} a^{2} + \frac{490390}{167959} a + \frac{49039}{15269} \) (order $22$)
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:  \( 26746667.8364 \) (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

\(\Q(\sqrt{-43}) \), \(\Q(\sqrt{473}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{-43})\), \(\Q(\zeta_{11})^+\), 10.0.31512565339032283.3, 10.10.346638218729355113.1, \(\Q(\zeta_{11})\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$