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Properties

Label 20.0.23032797693...4169.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 23^{10}$
Root discriminant $41.51$
Ramified primes $11, 23$
Class number $48$ (GRH)
Class group $[2, 2, 2, 6]$ (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![60466176, -10077696, -8398080, 3079296, 886464, -660960, -37584, 116424, -13140, -17214, 5059, -2869, -365, 539, -29, -85, 19, 11, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 11*x^17 + 19*x^16 - 85*x^15 - 29*x^14 + 539*x^13 - 365*x^12 - 2869*x^11 + 5059*x^10 - 17214*x^9 - 13140*x^8 + 116424*x^7 - 37584*x^6 - 660960*x^5 + 886464*x^4 + 3079296*x^3 - 8398080*x^2 - 10077696*x + 60466176)
 
gp: K = bnfinit(x^20 - x^19 - 5*x^18 + 11*x^17 + 19*x^16 - 85*x^15 - 29*x^14 + 539*x^13 - 365*x^12 - 2869*x^11 + 5059*x^10 - 17214*x^9 - 13140*x^8 + 116424*x^7 - 37584*x^6 - 660960*x^5 + 886464*x^4 + 3079296*x^3 - 8398080*x^2 - 10077696*x + 60466176, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 5 x^{18} + 11 x^{17} + 19 x^{16} - 85 x^{15} - 29 x^{14} + 539 x^{13} - 365 x^{12} - 2869 x^{11} + 5059 x^{10} - 17214 x^{9} - 13140 x^{8} + 116424 x^{7} - 37584 x^{6} - 660960 x^{5} + 886464 x^{4} + 3079296 x^{3} - 8398080 x^{2} - 10077696 x + 60466176 \)

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:  \(230327976934347149972494554684169=11^{18}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.51$
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:  $11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(253=11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{253}(1,·)$, $\chi_{253}(68,·)$, $\chi_{253}(70,·)$, $\chi_{253}(137,·)$, $\chi_{253}(139,·)$, $\chi_{253}(206,·)$, $\chi_{253}(208,·)$, $\chi_{253}(24,·)$, $\chi_{253}(91,·)$, $\chi_{253}(93,·)$, $\chi_{253}(160,·)$, $\chi_{253}(162,·)$, $\chi_{253}(229,·)$, $\chi_{253}(45,·)$, $\chi_{253}(47,·)$, $\chi_{253}(114,·)$, $\chi_{253}(116,·)$, $\chi_{253}(183,·)$, $\chi_{253}(185,·)$, $\chi_{253}(252,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{30354} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{2190}{5059}$, $\frac{1}{182124} a^{12} - \frac{1}{182124} a^{11} + \frac{13}{36} a^{10} - \frac{7}{36} a^{9} + \frac{1}{36} a^{8} + \frac{5}{36} a^{7} - \frac{11}{36} a^{6} + \frac{17}{36} a^{5} + \frac{13}{36} a^{4} - \frac{7}{36} a^{3} + \frac{1}{36} a^{2} - \frac{2869}{30354} a - \frac{365}{5059}$, $\frac{1}{1092744} a^{13} - \frac{1}{1092744} a^{12} - \frac{5}{1092744} a^{11} - \frac{7}{216} a^{10} - \frac{71}{216} a^{9} - \frac{103}{216} a^{8} + \frac{97}{216} a^{7} + \frac{89}{216} a^{6} - \frac{23}{216} a^{5} - \frac{79}{216} a^{4} + \frac{1}{216} a^{3} - \frac{2869}{182124} a^{2} - \frac{365}{30354} a + \frac{539}{5059}$, $\frac{1}{6556464} a^{14} - \frac{1}{6556464} a^{13} - \frac{5}{6556464} a^{12} + \frac{11}{6556464} a^{11} + \frac{145}{1296} a^{10} - \frac{103}{1296} a^{9} + \frac{529}{1296} a^{8} + \frac{89}{1296} a^{7} + \frac{625}{1296} a^{6} + \frac{137}{1296} a^{5} + \frac{1}{1296} a^{4} - \frac{2869}{1092744} a^{3} - \frac{365}{182124} a^{2} + \frac{539}{30354} a - \frac{29}{5059}$, $\frac{1}{39338784} a^{15} - \frac{1}{39338784} a^{14} - \frac{5}{39338784} a^{13} + \frac{11}{39338784} a^{12} + \frac{19}{39338784} a^{11} - \frac{103}{7776} a^{10} - \frac{767}{7776} a^{9} + \frac{1385}{7776} a^{8} + \frac{3217}{7776} a^{7} - \frac{3751}{7776} a^{6} + \frac{1}{7776} a^{5} - \frac{2869}{6556464} a^{4} - \frac{365}{1092744} a^{3} + \frac{539}{182124} a^{2} - \frac{29}{30354} a - \frac{85}{5059}$, $\frac{1}{236032704} a^{16} - \frac{1}{236032704} a^{15} - \frac{5}{236032704} a^{14} + \frac{11}{236032704} a^{13} + \frac{19}{236032704} a^{12} - \frac{85}{236032704} a^{11} + \frac{7009}{46656} a^{10} - \frac{6391}{46656} a^{9} + \frac{10993}{46656} a^{8} - \frac{19303}{46656} a^{7} + \frac{1}{46656} a^{6} - \frac{2869}{39338784} a^{5} - \frac{365}{6556464} a^{4} + \frac{539}{1092744} a^{3} - \frac{29}{182124} a^{2} - \frac{85}{30354} a + \frac{19}{5059}$, $\frac{1}{1416196224} a^{17} - \frac{1}{1416196224} a^{16} - \frac{5}{1416196224} a^{15} + \frac{11}{1416196224} a^{14} + \frac{19}{1416196224} a^{13} - \frac{85}{1416196224} a^{12} - \frac{29}{1416196224} a^{11} + \frac{133577}{279936} a^{10} + \frac{104305}{279936} a^{9} - \frac{65959}{279936} a^{8} + \frac{1}{279936} a^{7} - \frac{2869}{236032704} a^{6} - \frac{365}{39338784} a^{5} + \frac{539}{6556464} a^{4} - \frac{29}{1092744} a^{3} - \frac{85}{182124} a^{2} + \frac{19}{30354} a + \frac{11}{5059}$, $\frac{1}{8497177344} a^{18} - \frac{1}{8497177344} a^{17} - \frac{5}{8497177344} a^{16} + \frac{11}{8497177344} a^{15} + \frac{19}{8497177344} a^{14} - \frac{85}{8497177344} a^{13} - \frac{29}{8497177344} a^{12} + \frac{539}{8497177344} a^{11} - \frac{175631}{1679616} a^{10} - \frac{625831}{1679616} a^{9} + \frac{1}{1679616} a^{8} - \frac{2869}{1416196224} a^{7} - \frac{365}{236032704} a^{6} + \frac{539}{39338784} a^{5} - \frac{29}{6556464} a^{4} - \frac{85}{1092744} a^{3} + \frac{19}{182124} a^{2} + \frac{11}{30354} a - \frac{5}{5059}$, $\frac{1}{50983064064} a^{19} - \frac{1}{50983064064} a^{18} - \frac{5}{50983064064} a^{17} + \frac{11}{50983064064} a^{16} + \frac{19}{50983064064} a^{15} - \frac{85}{50983064064} a^{14} - \frac{29}{50983064064} a^{13} + \frac{539}{50983064064} a^{12} - \frac{365}{50983064064} a^{11} + \frac{1053785}{10077696} a^{10} + \frac{1}{10077696} a^{9} - \frac{2869}{8497177344} a^{8} - \frac{365}{1416196224} a^{7} + \frac{539}{236032704} a^{6} - \frac{29}{39338784} a^{5} - \frac{85}{6556464} a^{4} + \frac{19}{1092744} a^{3} + \frac{11}{182124} a^{2} - \frac{5}{30354} a - \frac{1}{5059}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (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:  \( -\frac{85}{236032704} a^{17} + \frac{1609901}{236032704} a^{6} \) (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:  \( 13106437.7678 \) (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

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{253}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{-23})\), \(\Q(\zeta_{11})^+\), 10.0.1379687283212183.1, 10.10.15176560115334013.1, \(\Q(\zeta_{11})\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
11Data not computed
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$