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Properties

Label 20.10.1453325601...0000.6
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{8}\cdot 5^{22}\cdot 47^{8}$
Root discriminant $36.15$
Ramified primes $2, 5, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, 0, 100, 0, 1500, 0, -1900, 0, -3125, 0, 4780, 0, -1160, 0, -410, 0, 75, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 + 75*x^16 - 410*x^14 - 1160*x^12 + 4780*x^10 - 3125*x^8 - 1900*x^6 + 1500*x^4 + 100*x^2 - 25)
 
gp: K = bnfinit(x^20 + 20*x^18 + 75*x^16 - 410*x^14 - 1160*x^12 + 4780*x^10 - 3125*x^8 - 1900*x^6 + 1500*x^4 + 100*x^2 - 25, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} + 75 x^{16} - 410 x^{14} - 1160 x^{12} + 4780 x^{10} - 3125 x^{8} - 1900 x^{6} + 1500 x^{4} + 100 x^{2} - 25 \)

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:  \(-14533256019141235351562500000000=-\,2^{8}\cdot 5^{22}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.15$
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:  $2, 5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{15} - \frac{1}{20} a^{13} - \frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{220} a^{16} - \frac{1}{55} a^{14} - \frac{1}{20} a^{13} - \frac{3}{110} a^{12} - \frac{9}{220} a^{10} - \frac{1}{4} a^{9} + \frac{3}{22} a^{8} + \frac{1}{4} a^{7} + \frac{1}{22} a^{6} - \frac{1}{4} a^{5} - \frac{2}{11} a^{4} + \frac{1}{4} a^{3} + \frac{19}{44} a^{2} + \frac{1}{4} a - \frac{5}{44}$, $\frac{1}{1100} a^{17} - \frac{3}{220} a^{15} + \frac{1}{220} a^{13} + \frac{7}{220} a^{11} - \frac{1}{20} a^{10} + \frac{39}{220} a^{9} - \frac{1}{4} a^{8} - \frac{21}{110} a^{7} - \frac{1}{2} a^{6} + \frac{5}{44} a^{5} - \frac{1}{4} a^{4} + \frac{17}{44} a^{3} - \frac{1}{2} a^{2} - \frac{1}{44} a - \frac{1}{4}$, $\frac{1}{48166397400} a^{18} - \frac{7072503}{3211093160} a^{16} - \frac{6032379}{401386645} a^{14} - \frac{1}{20} a^{13} + \frac{104046793}{4816639740} a^{12} + \frac{44402419}{2408319870} a^{10} - \frac{1}{4} a^{9} - \frac{356082997}{1605546580} a^{8} + \frac{1}{4} a^{7} + \frac{737682529}{1926655896} a^{6} - \frac{1}{4} a^{5} + \frac{182977}{58383512} a^{4} + \frac{1}{4} a^{3} - \frac{8549147}{642218632} a^{2} + \frac{1}{4} a + \frac{533122699}{1926655896}$, $\frac{1}{96332794800} a^{19} - \frac{1}{96332794800} a^{18} + \frac{8425119}{32110931600} a^{17} - \frac{1504675}{1284437264} a^{16} - \frac{4766819}{1605546580} a^{15} + \frac{6665159}{401386645} a^{14} - \frac{71103743}{9633279480} a^{13} + \frac{27316109}{9633279480} a^{12} + \frac{66911021}{9633279480} a^{11} + \frac{21647903}{1926655896} a^{10} + \frac{96389221}{3211093160} a^{9} + \frac{137144827}{3211093160} a^{8} + \frac{7804450241}{19266558960} a^{7} + \frac{1101398099}{3853311792} a^{6} + \frac{5490569}{116767024} a^{5} + \frac{10432207}{116767024} a^{4} + \frac{414731315}{1284437264} a^{3} - \frac{268772535}{1284437264} a^{2} + \frac{1365087745}{3853311792} a - \frac{314184529}{3853311792}$

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:  \( 290758999.075 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.6903125.1, 10.10.238265673828125.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
5Data not computed
$47$47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$