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Properties

Label 20.0.32061284906...5625.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $33.52$
Ramified primes $3, 5, 11$
Class number $44$ (GRH)
Class group $[2, 22]$ (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![1048576, -262144, -196608, 114688, 20480, -33792, 3328, 7616, -2736, -1220, 989, -305, -171, 119, 13, -33, 5, 7, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 3*x^18 + 7*x^17 + 5*x^16 - 33*x^15 + 13*x^14 + 119*x^13 - 171*x^12 - 305*x^11 + 989*x^10 - 1220*x^9 - 2736*x^8 + 7616*x^7 + 3328*x^6 - 33792*x^5 + 20480*x^4 + 114688*x^3 - 196608*x^2 - 262144*x + 1048576)
 
gp: K = bnfinit(x^20 - x^19 - 3*x^18 + 7*x^17 + 5*x^16 - 33*x^15 + 13*x^14 + 119*x^13 - 171*x^12 - 305*x^11 + 989*x^10 - 1220*x^9 - 2736*x^8 + 7616*x^7 + 3328*x^6 - 33792*x^5 + 20480*x^4 + 114688*x^3 - 196608*x^2 - 262144*x + 1048576, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 3 x^{18} + 7 x^{17} + 5 x^{16} - 33 x^{15} + 13 x^{14} + 119 x^{13} - 171 x^{12} - 305 x^{11} + 989 x^{10} - 1220 x^{9} - 2736 x^{8} + 7616 x^{7} + 3328 x^{6} - 33792 x^{5} + 20480 x^{4} + 114688 x^{3} - 196608 x^{2} - 262144 x + 1048576 \)

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:  \(3206128490667995866421572265625=3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.52$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(134,·)$, $\chi_{165}(136,·)$, $\chi_{165}(74,·)$, $\chi_{165}(76,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(149,·)$, $\chi_{165}(151,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(29,·)$, $\chi_{165}(31,·)$, $\chi_{165}(164,·)$, $\chi_{165}(104,·)$, $\chi_{165}(106,·)$, $\chi_{165}(46,·)$, $\chi_{165}(119,·)$, $\chi_{165}(59,·)$, $\chi_{165}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3956} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{305}{989}$, $\frac{1}{15824} a^{12} - \frac{1}{15824} a^{11} + \frac{1}{16} a^{10} + \frac{3}{16} a^{9} - \frac{7}{16} a^{8} - \frac{5}{16} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{7}{16} a^{4} - \frac{5}{16} a^{3} + \frac{1}{16} a^{2} - \frac{305}{3956} a - \frac{171}{989}$, $\frac{1}{63296} a^{13} - \frac{1}{63296} a^{12} - \frac{3}{63296} a^{11} - \frac{13}{64} a^{10} + \frac{9}{64} a^{9} - \frac{21}{64} a^{8} - \frac{15}{64} a^{7} - \frac{29}{64} a^{6} + \frac{25}{64} a^{5} + \frac{27}{64} a^{4} + \frac{1}{64} a^{3} - \frac{305}{15824} a^{2} - \frac{171}{3956} a + \frac{119}{989}$, $\frac{1}{253184} a^{14} - \frac{1}{253184} a^{13} - \frac{3}{253184} a^{12} + \frac{7}{253184} a^{11} + \frac{73}{256} a^{10} - \frac{21}{256} a^{9} - \frac{15}{256} a^{8} + \frac{99}{256} a^{7} - \frac{39}{256} a^{6} - \frac{101}{256} a^{5} + \frac{1}{256} a^{4} - \frac{305}{63296} a^{3} - \frac{171}{15824} a^{2} + \frac{119}{3956} a + \frac{13}{989}$, $\frac{1}{1012736} a^{15} - \frac{1}{1012736} a^{14} - \frac{3}{1012736} a^{13} + \frac{7}{1012736} a^{12} + \frac{5}{1012736} a^{11} + \frac{235}{1024} a^{10} + \frac{497}{1024} a^{9} - \frac{413}{1024} a^{8} + \frac{473}{1024} a^{7} + \frac{155}{1024} a^{6} + \frac{1}{1024} a^{5} - \frac{305}{253184} a^{4} - \frac{171}{63296} a^{3} + \frac{119}{15824} a^{2} + \frac{13}{3956} a - \frac{33}{989}$, $\frac{1}{4050944} a^{16} - \frac{1}{4050944} a^{15} - \frac{3}{4050944} a^{14} + \frac{7}{4050944} a^{13} + \frac{5}{4050944} a^{12} - \frac{33}{4050944} a^{11} + \frac{497}{4096} a^{10} - \frac{1437}{4096} a^{9} - \frac{551}{4096} a^{8} - \frac{1893}{4096} a^{7} + \frac{1}{4096} a^{6} - \frac{305}{1012736} a^{5} - \frac{171}{253184} a^{4} + \frac{119}{63296} a^{3} + \frac{13}{15824} a^{2} - \frac{33}{3956} a + \frac{5}{989}$, $\frac{1}{16203776} a^{17} - \frac{1}{16203776} a^{16} - \frac{3}{16203776} a^{15} + \frac{7}{16203776} a^{14} + \frac{5}{16203776} a^{13} - \frac{33}{16203776} a^{12} + \frac{13}{16203776} a^{11} - \frac{5533}{16384} a^{10} + \frac{3545}{16384} a^{9} + \frac{2203}{16384} a^{8} + \frac{1}{16384} a^{7} - \frac{305}{4050944} a^{6} - \frac{171}{1012736} a^{5} + \frac{119}{253184} a^{4} + \frac{13}{63296} a^{3} - \frac{33}{15824} a^{2} + \frac{5}{3956} a + \frac{7}{989}$, $\frac{1}{64815104} a^{18} - \frac{1}{64815104} a^{17} - \frac{3}{64815104} a^{16} + \frac{7}{64815104} a^{15} + \frac{5}{64815104} a^{14} - \frac{33}{64815104} a^{13} + \frac{13}{64815104} a^{12} + \frac{119}{64815104} a^{11} - \frac{29223}{65536} a^{10} - \frac{14181}{65536} a^{9} + \frac{1}{65536} a^{8} - \frac{305}{16203776} a^{7} - \frac{171}{4050944} a^{6} + \frac{119}{1012736} a^{5} + \frac{13}{253184} a^{4} - \frac{33}{63296} a^{3} + \frac{5}{15824} a^{2} + \frac{7}{3956} a - \frac{3}{989}$, $\frac{1}{259260416} a^{19} - \frac{1}{259260416} a^{18} - \frac{3}{259260416} a^{17} + \frac{7}{259260416} a^{16} + \frac{5}{259260416} a^{15} - \frac{33}{259260416} a^{14} + \frac{13}{259260416} a^{13} + \frac{119}{259260416} a^{12} - \frac{171}{259260416} a^{11} + \frac{116891}{262144} a^{10} + \frac{1}{262144} a^{9} - \frac{305}{64815104} a^{8} - \frac{171}{16203776} a^{7} + \frac{119}{4050944} a^{6} + \frac{13}{1012736} a^{5} - \frac{33}{253184} a^{4} + \frac{5}{63296} a^{3} + \frac{7}{15824} a^{2} - \frac{3}{3956} a - \frac{1}{989}$

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

Class group and class number

$C_{2}\times C_{22}$, which has order $44$ (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:  \( -\frac{305}{259260416} a^{19} + \frac{305}{259260416} a^{18} + \frac{915}{259260416} a^{17} - \frac{2135}{259260416} a^{16} - \frac{1525}{259260416} a^{15} + \frac{10065}{259260416} a^{14} - \frac{3965}{259260416} a^{13} - \frac{36295}{259260416} a^{12} + \frac{52155}{259260416} a^{11} - \frac{171}{262144} a^{10} - \frac{305}{262144} a^{9} + \frac{93025}{64815104} a^{8} + \frac{52155}{16203776} a^{7} - \frac{36295}{4050944} a^{6} - \frac{3965}{1012736} a^{5} + \frac{10065}{253184} a^{4} - \frac{1525}{63296} a^{3} - \frac{2135}{15824} a^{2} + \frac{915}{3956} a + \frac{305}{989} \) (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:  \( 2681477.98686 \) (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

\(\Q(\sqrt{165}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 10.10.1790566527853125.1, \(\Q(\zeta_{11})\), 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$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