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Properties

Label 20.0.14857952975...7201.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 47^{10}$
Root discriminant $18.14$
Ramified primes $7, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -512, -4096, 384, 8768, 160, -9840, -856, 7408, 322, -4157, 161, 1852, -107, -615, 5, 137, 3, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 16*x^18 + 3*x^17 + 137*x^16 + 5*x^15 - 615*x^14 - 107*x^13 + 1852*x^12 + 161*x^11 - 4157*x^10 + 322*x^9 + 7408*x^8 - 856*x^7 - 9840*x^6 + 160*x^5 + 8768*x^4 + 384*x^3 - 4096*x^2 - 512*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 - 16*x^18 + 3*x^17 + 137*x^16 + 5*x^15 - 615*x^14 - 107*x^13 + 1852*x^12 + 161*x^11 - 4157*x^10 + 322*x^9 + 7408*x^8 - 856*x^7 - 9840*x^6 + 160*x^5 + 8768*x^4 + 384*x^3 - 4096*x^2 - 512*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 16 x^{18} + 3 x^{17} + 137 x^{16} + 5 x^{15} - 615 x^{14} - 107 x^{13} + 1852 x^{12} + 161 x^{11} - 4157 x^{10} + 322 x^{9} + 7408 x^{8} - 856 x^{7} - 9840 x^{6} + 160 x^{5} + 8768 x^{4} + 384 x^{3} - 4096 x^{2} - 512 x + 1024 \)

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:  \(14857952975500019812957201=7^{10}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.14$
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:  $7, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{80} a^{14} + \frac{3}{80} a^{13} - \frac{1}{20} a^{12} - \frac{13}{80} a^{11} + \frac{1}{16} a^{10} + \frac{9}{80} a^{9} - \frac{19}{80} a^{8} + \frac{5}{16} a^{7} - \frac{2}{5} a^{6} - \frac{31}{80} a^{5} - \frac{5}{16} a^{4} - \frac{1}{40} a^{3} + \frac{1}{10} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{160} a^{15} - \frac{1}{160} a^{14} + \frac{1}{40} a^{13} - \frac{17}{160} a^{12} - \frac{23}{160} a^{11} - \frac{31}{160} a^{10} + \frac{9}{32} a^{9} - \frac{39}{160} a^{8} + \frac{3}{10} a^{7} + \frac{37}{160} a^{6} + \frac{19}{160} a^{5} - \frac{21}{80} a^{4} + \frac{19}{40} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{320} a^{16} - \frac{1}{320} a^{15} + \frac{11}{320} a^{13} + \frac{33}{320} a^{12} - \frac{59}{320} a^{11} - \frac{3}{64} a^{10} - \frac{23}{64} a^{9} - \frac{19}{80} a^{8} + \frac{57}{320} a^{7} + \frac{107}{320} a^{6} - \frac{79}{160} a^{5} + \frac{7}{40} a^{4} + \frac{1}{5} a^{3} - \frac{7}{20} a^{2} - \frac{2}{5}$, $\frac{1}{640} a^{17} - \frac{1}{640} a^{16} + \frac{3}{640} a^{14} + \frac{9}{640} a^{13} - \frac{27}{640} a^{12} + \frac{89}{640} a^{11} - \frac{31}{128} a^{10} - \frac{37}{160} a^{9} + \frac{209}{640} a^{8} + \frac{227}{640} a^{7} + \frac{49}{320} a^{6} - \frac{1}{40} a^{5} - \frac{7}{80} a^{4} - \frac{3}{20} a^{3} - \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{283021024000} a^{18} - \frac{174961663}{283021024000} a^{17} - \frac{119919781}{141510512000} a^{16} - \frac{289952877}{283021024000} a^{15} - \frac{12188041}{4354169600} a^{14} + \frac{17123161839}{283021024000} a^{13} - \frac{1044880653}{283021024000} a^{12} + \frac{20360215751}{283021024000} a^{11} - \frac{13028157277}{141510512000} a^{10} + \frac{40766088057}{283021024000} a^{9} - \frac{27569399683}{283021024000} a^{8} + \frac{5698106919}{17688814000} a^{7} + \frac{14935722169}{70755256000} a^{6} - \frac{3267813443}{8844407000} a^{5} - \frac{48187947}{272135600} a^{4} + \frac{525645121}{8844407000} a^{3} + \frac{543410744}{1105550875} a^{2} - \frac{174961663}{1105550875} a + \frac{442220352}{1105550875}$, $\frac{1}{566042048000} a^{19} - \frac{1}{566042048000} a^{18} + \frac{110555083}{141510512000} a^{17} + \frac{792143679}{566042048000} a^{16} - \frac{404761439}{566042048000} a^{15} - \frac{1631195991}{566042048000} a^{14} - \frac{4724423047}{113208409600} a^{13} + \frac{10751105173}{113208409600} a^{12} - \frac{1051221289}{4422203500} a^{11} - \frac{196824214891}{566042048000} a^{10} + \frac{136134491451}{566042048000} a^{9} + \frac{19596989879}{283021024000} a^{8} + \frac{22297443431}{141510512000} a^{7} + \frac{26294031067}{70755256000} a^{6} - \frac{6251755737}{35377628000} a^{5} + \frac{6535626891}{17688814000} a^{4} + \frac{1897773677}{8844407000} a^{3} - \frac{103670817}{442220350} a^{2} + \frac{903170071}{2211101750} a + \frac{174961662}{1105550875}$

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:  \( 18347.7720663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{329}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{-47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.2.3854601532649.1 x5, 10.0.82012798567.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$