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Properties

Label 20.0.15705390275...3769.2
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 11^{18}$
Root discriminant $22.90$
Ramified primes $7, 11$
Class number $5$
Class group $[5]$
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![1024, -512, -256, 384, -64, -160, 112, 24, -68, 22, 23, 11, -17, 3, 7, -5, -1, 3, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 3*x^17 - x^16 - 5*x^15 + 7*x^14 + 3*x^13 - 17*x^12 + 11*x^11 + 23*x^10 + 22*x^9 - 68*x^8 + 24*x^7 + 112*x^6 - 160*x^5 - 64*x^4 + 384*x^3 - 256*x^2 - 512*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 - x^18 + 3*x^17 - x^16 - 5*x^15 + 7*x^14 + 3*x^13 - 17*x^12 + 11*x^11 + 23*x^10 + 22*x^9 - 68*x^8 + 24*x^7 + 112*x^6 - 160*x^5 - 64*x^4 + 384*x^3 - 256*x^2 - 512*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} + 3 x^{17} - x^{16} - 5 x^{15} + 7 x^{14} + 3 x^{13} - 17 x^{12} + 11 x^{11} + 23 x^{10} + 22 x^{9} - 68 x^{8} + 24 x^{7} + 112 x^{6} - 160 x^{5} - 64 x^{4} + 384 x^{3} - 256 x^{2} - 512 x + 1024 \)

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:  \(1570539027548129147161113769=7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.90$
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:  $7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(77=7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(69,·)$, $\chi_{77}(6,·)$, $\chi_{77}(71,·)$, $\chi_{77}(8,·)$, $\chi_{77}(76,·)$, $\chi_{77}(13,·)$, $\chi_{77}(15,·)$, $\chi_{77}(20,·)$, $\chi_{77}(27,·)$, $\chi_{77}(29,·)$, $\chi_{77}(34,·)$, $\chi_{77}(36,·)$, $\chi_{77}(41,·)$, $\chi_{77}(43,·)$, $\chi_{77}(48,·)$, $\chi_{77}(50,·)$, $\chi_{77}(57,·)$, $\chi_{77}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{46} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{11}{23}$, $\frac{1}{92} a^{12} - \frac{1}{92} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{11}{46} a + \frac{6}{23}$, $\frac{1}{184} a^{13} - \frac{1}{184} a^{12} - \frac{1}{184} a^{11} - \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} + \frac{11}{92} a^{2} - \frac{17}{46} a + \frac{3}{23}$, $\frac{1}{368} a^{14} - \frac{1}{368} a^{13} - \frac{1}{368} a^{12} + \frac{3}{368} a^{11} - \frac{7}{16} a^{10} - \frac{3}{16} a^{9} + \frac{1}{16} a^{8} + \frac{5}{16} a^{7} - \frac{7}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{11}{184} a^{3} - \frac{17}{92} a^{2} + \frac{3}{46} a + \frac{7}{23}$, $\frac{1}{736} a^{15} - \frac{1}{736} a^{14} - \frac{1}{736} a^{13} + \frac{3}{736} a^{12} - \frac{1}{736} a^{11} - \frac{3}{32} a^{10} - \frac{15}{32} a^{9} - \frac{11}{32} a^{8} + \frac{9}{32} a^{7} + \frac{13}{32} a^{6} + \frac{1}{32} a^{5} + \frac{11}{368} a^{4} - \frac{17}{184} a^{3} + \frac{3}{92} a^{2} + \frac{7}{46} a - \frac{5}{23}$, $\frac{1}{1472} a^{16} - \frac{1}{1472} a^{15} - \frac{1}{1472} a^{14} + \frac{3}{1472} a^{13} - \frac{1}{1472} a^{12} - \frac{5}{1472} a^{11} + \frac{17}{64} a^{10} - \frac{11}{64} a^{9} - \frac{23}{64} a^{8} - \frac{19}{64} a^{7} + \frac{1}{64} a^{6} + \frac{11}{736} a^{5} - \frac{17}{368} a^{4} + \frac{3}{184} a^{3} + \frac{7}{92} a^{2} - \frac{5}{46} a - \frac{1}{23}$, $\frac{1}{2944} a^{17} - \frac{1}{2944} a^{16} - \frac{1}{2944} a^{15} + \frac{3}{2944} a^{14} - \frac{1}{2944} a^{13} - \frac{5}{2944} a^{12} + \frac{7}{2944} a^{11} - \frac{11}{128} a^{10} - \frac{23}{128} a^{9} + \frac{45}{128} a^{8} + \frac{1}{128} a^{7} + \frac{11}{1472} a^{6} - \frac{17}{736} a^{5} + \frac{3}{368} a^{4} + \frac{7}{184} a^{3} - \frac{5}{92} a^{2} - \frac{1}{46} a + \frac{3}{23}$, $\frac{1}{5888} a^{18} - \frac{1}{5888} a^{17} - \frac{1}{5888} a^{16} + \frac{3}{5888} a^{15} - \frac{1}{5888} a^{14} - \frac{5}{5888} a^{13} + \frac{7}{5888} a^{12} + \frac{3}{5888} a^{11} - \frac{23}{256} a^{10} + \frac{45}{256} a^{9} + \frac{1}{256} a^{8} + \frac{11}{2944} a^{7} - \frac{17}{1472} a^{6} + \frac{3}{736} a^{5} + \frac{7}{368} a^{4} - \frac{5}{184} a^{3} - \frac{1}{92} a^{2} + \frac{3}{46} a - \frac{1}{23}$, $\frac{1}{11776} a^{19} - \frac{1}{11776} a^{18} - \frac{1}{11776} a^{17} + \frac{3}{11776} a^{16} - \frac{1}{11776} a^{15} - \frac{5}{11776} a^{14} + \frac{7}{11776} a^{13} + \frac{3}{11776} a^{12} - \frac{17}{11776} a^{11} + \frac{45}{512} a^{10} + \frac{1}{512} a^{9} + \frac{11}{5888} a^{8} - \frac{17}{2944} a^{7} + \frac{3}{1472} a^{6} + \frac{7}{736} a^{5} - \frac{5}{368} a^{4} - \frac{1}{184} a^{3} + \frac{3}{92} a^{2} - \frac{1}{46} a - \frac{1}{23}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( \frac{1}{46} a^{12} + \frac{45}{46} a \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 707570.081247 \)
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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, \(\Q(\zeta_{11})\), 10.0.3602729712967.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
7Data not computed
$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}$