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Properties

Label 20.10.1092960334...9968.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{40}\cdot 1583^{5}$
Root discriminant $25.23$
Ramified primes $2, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, -232, 876, -1444, 580, 1388, -1644, -327, 1500, -962, 208, 215, -368, 260, -36, -71, 44, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 4*x^18 + 44*x^17 - 71*x^16 - 36*x^15 + 260*x^14 - 368*x^13 + 215*x^12 + 208*x^11 - 962*x^10 + 1500*x^9 - 327*x^8 - 1644*x^7 + 1388*x^6 + 580*x^5 - 1444*x^4 + 876*x^3 - 232*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 4*x^18 + 44*x^17 - 71*x^16 - 36*x^15 + 260*x^14 - 368*x^13 + 215*x^12 + 208*x^11 - 962*x^10 + 1500*x^9 - 327*x^8 - 1644*x^7 + 1388*x^6 + 580*x^5 - 1444*x^4 + 876*x^3 - 232*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 4 x^{18} + 44 x^{17} - 71 x^{16} - 36 x^{15} + 260 x^{14} - 368 x^{13} + 215 x^{12} + 208 x^{11} - 962 x^{10} + 1500 x^{9} - 327 x^{8} - 1644 x^{7} + 1388 x^{6} + 580 x^{5} - 1444 x^{4} + 876 x^{3} - 232 x^{2} + 20 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10929603344110461118702419968=-\,2^{40}\cdot 1583^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.23$
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:  $2, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2404984661722221019915621} a^{19} + \frac{561972496876361468426450}{2404984661722221019915621} a^{18} + \frac{398999082698681078437102}{2404984661722221019915621} a^{17} + \frac{86301261475476101577729}{343569237388888717130803} a^{16} + \frac{402739489416549529133175}{2404984661722221019915621} a^{15} - \frac{685618308871252442588843}{2404984661722221019915621} a^{14} + \frac{310814233895394213630928}{2404984661722221019915621} a^{13} + \frac{93609438989298466520755}{343569237388888717130803} a^{12} + \frac{49896309819162483354116}{2404984661722221019915621} a^{11} - \frac{36156051323425524385971}{2404984661722221019915621} a^{10} - \frac{1152920717076349862227945}{2404984661722221019915621} a^{9} - \frac{149110423965470899850867}{343569237388888717130803} a^{8} - \frac{72322164247164391116338}{2404984661722221019915621} a^{7} + \frac{836566866822532134417861}{2404984661722221019915621} a^{6} - \frac{1100853721121198551886136}{2404984661722221019915621} a^{5} - \frac{167809921247149503312807}{2404984661722221019915621} a^{4} + \frac{580795192339858160624878}{2404984661722221019915621} a^{3} + \frac{1130773928487184339754863}{2404984661722221019915621} a^{2} + \frac{829504912919704301794796}{2404984661722221019915621} a + \frac{1103843927106814906635794}{2404984661722221019915621}$

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:  $14$
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:  \( 3666885.92702 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.82112970752.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
1583Data not computed