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Properties

Label 20.0.20368198195...0961.6
Degree $20$
Signature $[0, 10]$
Discriminant $11^{16}\cdot 1451^{4}$
Root discriminant $29.20$
Ramified primes $11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T314

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![859, -331, -1019, -1591, 511, -130, 1130, 1351, 1700, 192, 513, -157, 56, -103, -1, -31, 15, -2, 8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 8*x^18 - 2*x^17 + 15*x^16 - 31*x^15 - x^14 - 103*x^13 + 56*x^12 - 157*x^11 + 513*x^10 + 192*x^9 + 1700*x^8 + 1351*x^7 + 1130*x^6 - 130*x^5 + 511*x^4 - 1591*x^3 - 1019*x^2 - 331*x + 859)
 
gp: K = bnfinit(x^20 - 2*x^19 + 8*x^18 - 2*x^17 + 15*x^16 - 31*x^15 - x^14 - 103*x^13 + 56*x^12 - 157*x^11 + 513*x^10 + 192*x^9 + 1700*x^8 + 1351*x^7 + 1130*x^6 - 130*x^5 + 511*x^4 - 1591*x^3 - 1019*x^2 - 331*x + 859, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 8 x^{18} - 2 x^{17} + 15 x^{16} - 31 x^{15} - x^{14} - 103 x^{13} + 56 x^{12} - 157 x^{11} + 513 x^{10} + 192 x^{9} + 1700 x^{8} + 1351 x^{7} + 1130 x^{6} - 130 x^{5} + 511 x^{4} - 1591 x^{3} - 1019 x^{2} - 331 x + 859 \)

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:  \(203681981950950327645213870961=11^{16}\cdot 1451^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.20$
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:  $11, 1451$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{79022} a^{18} - \frac{8969}{39511} a^{17} + \frac{12379}{79022} a^{16} + \frac{8939}{39511} a^{15} + \frac{13751}{39511} a^{14} + \frac{11905}{39511} a^{13} + \frac{12177}{39511} a^{12} + \frac{19638}{39511} a^{11} - \frac{10405}{39511} a^{10} - \frac{8164}{39511} a^{9} + \frac{14902}{39511} a^{8} + \frac{5076}{39511} a^{7} + \frac{4211}{79022} a^{6} + \frac{18415}{39511} a^{5} + \frac{35133}{79022} a^{4} + \frac{4967}{79022} a^{3} - \frac{30091}{79022} a^{2} - \frac{17403}{79022} a + \frac{6874}{39511}$, $\frac{1}{90829839140299270170106191987598} a^{19} - \frac{200422935847590296756686901}{90829839140299270170106191987598} a^{18} + \frac{1356269762073101644738087264041}{45414919570149635085053095993799} a^{17} + \frac{1156514735211332918954876114943}{45414919570149635085053095993799} a^{16} + \frac{3960695314990180514767969918383}{45414919570149635085053095993799} a^{15} - \frac{3788059046067964799512562458633}{90829839140299270170106191987598} a^{14} - \frac{9311418740608267025198163889711}{45414919570149635085053095993799} a^{13} - \frac{33718664045710776218590473451319}{90829839140299270170106191987598} a^{12} - \frac{21690998964781462429757431464623}{90829839140299270170106191987598} a^{11} - \frac{2574000210875558220638721634533}{45414919570149635085053095993799} a^{10} + \frac{42415027992489866715553979436261}{90829839140299270170106191987598} a^{9} - \frac{13842606020381862770043458583333}{90829839140299270170106191987598} a^{8} + \frac{41058914176326355515873029053879}{90829839140299270170106191987598} a^{7} + \frac{15987075667150129476300483142354}{45414919570149635085053095993799} a^{6} + \frac{4783011599968579267014712560329}{90829839140299270170106191987598} a^{5} - \frac{22325100234726973710817341313069}{90829839140299270170106191987598} a^{4} - \frac{25230137978526685724164525567579}{90829839140299270170106191987598} a^{3} + \frac{2313839262149421986398941186757}{90829839140299270170106191987598} a^{2} - \frac{39473615150587390059414409852927}{90829839140299270170106191987598} a + \frac{12792507578092002743970822903385}{90829839140299270170106191987598}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2063292.90078 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T314:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n314
Character table for t20n314 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.311034736331.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
1451Data not computed