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Properties

Label 20.0.750...776.1
Degree $20$
Signature $[0, 10]$
Discriminant $7.507\times 10^{20}$
Root discriminant $11.06$
Ramified primes $2, 41381$
Class number $1$
Class group trivial
Galois group 20T673

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(x^20 + x^16 + 5*x^12 + 10*x^10 + 9*x^8 + 7*x^6 + 3*x^4 + 4*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 4, 0, 3, 0, 7, 0, 9, 0, 10, 0, 5, 0, 0, 0, 1, 0, 0, 0, 1]);
 

\( x^{20} + x^{16} + 5 x^{12} + 10 x^{10} + 9 x^{8} + 7 x^{6} + 3 x^{4} + 4 x^{2} + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(750661066024355819776\)\(\medspace = 2^{8}\cdot 41381^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.06$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 41381$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{8326} a^{18} + \frac{1123}{8326} a^{16} - \frac{259}{8326} a^{14} - \frac{1805}{4163} a^{12} - \frac{1}{2} a^{11} + \frac{737}{8326} a^{10} + \frac{3387}{8326} a^{8} - \frac{686}{4163} a^{6} - \frac{1}{2} a^{5} + \frac{1862}{4163} a^{4} - \frac{1}{2} a^{3} - \frac{880}{4163} a^{2} + \frac{949}{8326}$, $\frac{1}{8326} a^{19} + \frac{1123}{8326} a^{17} - \frac{259}{8326} a^{15} - \frac{1805}{4163} a^{13} - \frac{1}{2} a^{12} + \frac{737}{8326} a^{11} + \frac{3387}{8326} a^{9} - \frac{686}{4163} a^{7} - \frac{1}{2} a^{6} + \frac{1862}{4163} a^{5} - \frac{1}{2} a^{4} - \frac{880}{4163} a^{3} + \frac{949}{8326} a$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 91.2702526158 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 91.2702526158 \cdot 1}{2\sqrt{750661066024355819776}}\approx 0.159726139411$

Galois group

20T673:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 61440
The 126 conjugacy class representatives for t20n673 are not computed
Character table for t20n673 is not computed

Intermediate fields

5.1.41381.1, 10.2.1712387161.1

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 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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
41381Data not computed