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Properties

Label 20.4.22195602257...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{46}\cdot 5^{12}\cdot 7^{5}\cdot 7687$
Root discriminant $32.91$
Ramified primes $2, 5, 7, 7687$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -200, -2250, -1360, -3775, -1512, -3100, -480, -1010, -40, -250, 0, 115, 0, 120, 0, 50, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 + 50*x^16 + 120*x^14 + 115*x^12 - 250*x^10 - 40*x^9 - 1010*x^8 - 480*x^7 - 3100*x^6 - 1512*x^5 - 3775*x^4 - 1360*x^3 - 2250*x^2 - 200*x + 8)
 
gp: K = bnfinit(x^20 + 10*x^18 + 50*x^16 + 120*x^14 + 115*x^12 - 250*x^10 - 40*x^9 - 1010*x^8 - 480*x^7 - 3100*x^6 - 1512*x^5 - 3775*x^4 - 1360*x^3 - 2250*x^2 - 200*x + 8, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} + 50 x^{16} + 120 x^{14} + 115 x^{12} - 250 x^{10} - 40 x^{9} - 1010 x^{8} - 480 x^{7} - 3100 x^{6} - 1512 x^{5} - 3775 x^{4} - 1360 x^{3} - 2250 x^{2} - 200 x + 8 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2219560225793376256000000000000=2^{46}\cdot 5^{12}\cdot 7^{5}\cdot 7687\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.91$
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:  $2, 5, 7, 7687$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} - \frac{2}{5} a^{9} - \frac{7}{20} a^{8} - \frac{1}{10} a^{7} + \frac{7}{20} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{11} + \frac{1}{10} a^{10} - \frac{3}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{20} a^{7} - \frac{1}{2} a^{6} + \frac{7}{20} a^{5} + \frac{1}{20} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{11} - \frac{3}{20} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{7}{20} a^{7} + \frac{7}{20} a^{6} - \frac{3}{10} a^{4} - \frac{7}{20} a^{3} + \frac{1}{5} a^{2} + \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{20} a^{15} + \frac{1}{10} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{5}$, $\frac{1}{20} a^{16} + \frac{1}{10} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a$, $\frac{1}{20} a^{17} - \frac{1}{20} a^{11} + \frac{3}{10} a^{9} + \frac{1}{5} a^{8} - \frac{7}{20} a^{7} + \frac{1}{4} a^{5} - \frac{1}{5} a^{4} + \frac{3}{20} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{100} a^{18} - \frac{1}{100} a^{17} + \frac{1}{50} a^{16} + \frac{1}{50} a^{15} + \frac{1}{50} a^{13} - \frac{1}{50} a^{12} - \frac{3}{50} a^{11} + \frac{1}{25} a^{10} + \frac{1}{5} a^{9} - \frac{9}{25} a^{8} - \frac{1}{25} a^{7} + \frac{23}{100} a^{6} + \frac{33}{100} a^{5} - \frac{1}{2} a^{4} - \frac{17}{50} a^{3} - \frac{23}{50} a^{2} - \frac{2}{25} a - \frac{12}{25}$, $\frac{1}{831440887998339234000} a^{19} - \frac{1912443032342449139}{415720443999169617000} a^{18} + \frac{107232030521820011}{5398966805184021000} a^{17} - \frac{13743530942072626}{674870850648002625} a^{16} + \frac{6648655181399240573}{415720443999169617000} a^{15} + \frac{1592368777928150603}{207860221999584808500} a^{14} + \frac{481539718579644023}{103930110999792404250} a^{13} + \frac{733240512789092431}{103930110999792404250} a^{12} + \frac{85454643665863978571}{831440887998339234000} a^{11} + \frac{84030531207034657481}{415720443999169617000} a^{10} + \frac{206435308207234035557}{415720443999169617000} a^{9} - \frac{49379518776500638433}{207860221999584808500} a^{8} + \frac{29664334268813076981}{138573481333056539000} a^{7} + \frac{13790303061927346763}{34643370333264134750} a^{6} + \frac{73689774048148398941}{207860221999584808500} a^{5} + \frac{21337567953883293799}{207860221999584808500} a^{4} - \frac{180503126971373810663}{831440887998339234000} a^{3} + \frac{7419944733339843457}{37792767636288147000} a^{2} - \frac{96864635459917490531}{415720443999169617000} a + \frac{34269496069143272309}{207860221999584808500}$

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:  $11$
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:  \( 134905553.929 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.2.6422528000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7687Data not computed