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Properties

Label 20.8.11348934571...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 3409004107^{2}$
Root discriminant $20.08$
Ramified primes $5, 3409004107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -9, 4, -30, 3, 80, -59, 119, -82, -134, 232, -29, 40, 56, -32, -6, -4, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^18 - 4*x^17 - 6*x^16 - 32*x^15 + 56*x^14 + 40*x^13 - 29*x^12 + 232*x^11 - 134*x^10 - 82*x^9 + 119*x^8 - 59*x^7 + 80*x^6 + 3*x^5 - 30*x^4 + 4*x^3 - 9*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^18 - 4*x^17 - 6*x^16 - 32*x^15 + 56*x^14 + 40*x^13 - 29*x^12 + 232*x^11 - 134*x^10 - 82*x^9 + 119*x^8 - 59*x^7 + 80*x^6 + 3*x^5 - 30*x^4 + 4*x^3 - 9*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{18} - 4 x^{17} - 6 x^{16} - 32 x^{15} + 56 x^{14} + 40 x^{13} - 29 x^{12} + 232 x^{11} - 134 x^{10} - 82 x^{9} + 119 x^{8} - 59 x^{7} + 80 x^{6} + 3 x^{5} - 30 x^{4} + 4 x^{3} - 9 x^{2} + 3 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(113489345718192064931640625=5^{10}\cdot 3409004107^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.08$
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:  $5, 3409004107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{8}{43} a^{17} - \frac{4}{43} a^{16} + \frac{3}{43} a^{15} + \frac{20}{43} a^{14} + \frac{19}{43} a^{13} - \frac{17}{43} a^{12} + \frac{2}{43} a^{11} + \frac{17}{43} a^{10} - \frac{20}{43} a^{9} + \frac{20}{43} a^{8} + \frac{2}{43} a^{7} - \frac{4}{43} a^{6} - \frac{6}{43} a^{5} - \frac{9}{43} a^{4} + \frac{20}{43} a^{3} - \frac{16}{43} a^{2} - \frac{5}{43} a + \frac{2}{43}$, $\frac{1}{108436391195777884871} a^{19} - \frac{1097606318157935968}{108436391195777884871} a^{18} - \frac{8851547677722651861}{108436391195777884871} a^{17} - \frac{39242219850338923280}{108436391195777884871} a^{16} + \frac{23968629284607255851}{108436391195777884871} a^{15} + \frac{9934176801115497437}{108436391195777884871} a^{14} + \frac{15081224643316631237}{108436391195777884871} a^{13} - \frac{33569355435852363356}{108436391195777884871} a^{12} - \frac{20528096993099659278}{108436391195777884871} a^{11} - \frac{27146201409913236618}{108436391195777884871} a^{10} - \frac{15199338687990333280}{108436391195777884871} a^{9} - \frac{28296699228353309453}{108436391195777884871} a^{8} - \frac{7817241319044604271}{108436391195777884871} a^{7} + \frac{49085123594459800949}{108436391195777884871} a^{6} - \frac{37878563995231616630}{108436391195777884871} a^{5} - \frac{20118554126782758492}{108436391195777884871} a^{4} + \frac{29139533521119968352}{108436391195777884871} a^{3} + \frac{4462377661990932016}{108436391195777884871} a^{2} - \frac{315174714630156265}{108436391195777884871} a + \frac{2200041797208819971}{108436391195777884871}$

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:  $13$
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:  \( 228494.705934 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.3409004107.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3409004107Data not computed