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Properties

Label 20.8.14282281564...5941.1
Degree $20$
Signature $[8, 6]$
Discriminant $7^{2}\cdot 13^{10}\cdot 61\cdot 277^{8}$
Root discriminant $51.02$
Ramified primes $7, 13, 61, 277$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, 67, -162, 74, 727, -2545, 4576, -5050, 2866, 1069, -4540, 5909, -5160, 3414, -1776, 732, -237, 58, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 58*x^18 - 237*x^17 + 732*x^16 - 1776*x^15 + 3414*x^14 - 5160*x^13 + 5909*x^12 - 4540*x^11 + 1069*x^10 + 2866*x^9 - 5050*x^8 + 4576*x^7 - 2545*x^6 + 727*x^5 + 74*x^4 - 162*x^3 + 67*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 58*x^18 - 237*x^17 + 732*x^16 - 1776*x^15 + 3414*x^14 - 5160*x^13 + 5909*x^12 - 4540*x^11 + 1069*x^10 + 2866*x^9 - 5050*x^8 + 4576*x^7 - 2545*x^6 + 727*x^5 + 74*x^4 - 162*x^3 + 67*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 58 x^{18} - 237 x^{17} + 732 x^{16} - 1776 x^{15} + 3414 x^{14} - 5160 x^{13} + 5909 x^{12} - 4540 x^{11} + 1069 x^{10} + 2866 x^{9} - 5050 x^{8} + 4576 x^{7} - 2545 x^{6} + 727 x^{5} + 74 x^{4} - 162 x^{3} + 67 x^{2} - 13 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14282281564801524093709347890645941=7^{2}\cdot 13^{10}\cdot 61\cdot 277^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.02$
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:  $7, 13, 61, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{15} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{7} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{133} a^{18} - \frac{9}{133} a^{17} - \frac{1}{19} a^{16} - \frac{6}{133} a^{15} - \frac{3}{133} a^{14} - \frac{8}{19} a^{13} - \frac{65}{133} a^{12} + \frac{39}{133} a^{11} + \frac{50}{133} a^{10} + \frac{2}{19} a^{9} - \frac{64}{133} a^{8} + \frac{23}{133} a^{7} + \frac{20}{133} a^{6} - \frac{17}{133} a^{5} - \frac{34}{133} a^{4} - \frac{8}{133} a^{3} + \frac{32}{133} a^{2} - \frac{43}{133} a - \frac{1}{133}$, $\frac{1}{133} a^{19} + \frac{1}{19} a^{17} + \frac{1}{19} a^{16} + \frac{2}{7} a^{15} + \frac{50}{133} a^{14} - \frac{8}{19} a^{13} - \frac{2}{19} a^{12} + \frac{59}{133} a^{11} + \frac{27}{133} a^{10} + \frac{62}{133} a^{9} - \frac{3}{19} a^{8} - \frac{1}{133} a^{7} - \frac{46}{133} a^{6} + \frac{22}{133} a^{5} + \frac{4}{19} a^{4} - \frac{40}{133} a^{3} - \frac{2}{133} a^{2} - \frac{27}{133} a + \frac{29}{133}$

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:  $13$
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:  \( 4292506348.85 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.12967201.1, 10.10.2185927923067213.1

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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
277Data not computed