↑

Properties

Label 20.0.83644670557...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{11}\cdot 31\cdot 1459^{4}\cdot 121951$
Root discriminant $22.19$
Ramified primes $5, 31, 1459, 121951$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![251, -1843, 7146, -19036, 39041, -64742, 90006, -106721, 109636, -98188, 77195, -53273, 32320, -17137, 7913, -3136, 1055, -291, 64, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 64*x^18 - 291*x^17 + 1055*x^16 - 3136*x^15 + 7913*x^14 - 17137*x^13 + 32320*x^12 - 53273*x^11 + 77195*x^10 - 98188*x^9 + 109636*x^8 - 106721*x^7 + 90006*x^6 - 64742*x^5 + 39041*x^4 - 19036*x^3 + 7146*x^2 - 1843*x + 251)
 
gp: K = bnfinit(x^20 - 10*x^19 + 64*x^18 - 291*x^17 + 1055*x^16 - 3136*x^15 + 7913*x^14 - 17137*x^13 + 32320*x^12 - 53273*x^11 + 77195*x^10 - 98188*x^9 + 109636*x^8 - 106721*x^7 + 90006*x^6 - 64742*x^5 + 39041*x^4 - 19036*x^3 + 7146*x^2 - 1843*x + 251, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 64 x^{18} - 291 x^{17} + 1055 x^{16} - 3136 x^{15} + 7913 x^{14} - 17137 x^{13} + 32320 x^{12} - 53273 x^{11} + 77195 x^{10} - 98188 x^{9} + 109636 x^{8} - 106721 x^{7} + 90006 x^{6} - 64742 x^{5} + 39041 x^{4} - 19036 x^{3} + 7146 x^{2} - 1843 x + 251 \)

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:  \(836446705572425050830078125=5^{11}\cdot 31\cdot 1459^{4}\cdot 121951\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.19$
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:  $5, 31, 1459, 121951$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{8}{17} a^{17} - \frac{4}{17} a^{16} - \frac{2}{17} a^{15} - \frac{3}{17} a^{14} + \frac{5}{17} a^{13} + \frac{3}{17} a^{12} - \frac{4}{17} a^{11} - \frac{8}{17} a^{10} - \frac{5}{17} a^{9} + \frac{6}{17} a^{8} - \frac{1}{17} a^{7} + \frac{5}{17} a^{6} + \frac{1}{17} a^{5} + \frac{3}{17} a^{4} + \frac{6}{17} a^{3} + \frac{8}{17} a^{2} - \frac{2}{17} a + \frac{8}{17}$, $\frac{1}{17} a^{19} - \frac{4}{17} a^{16} - \frac{4}{17} a^{15} - \frac{5}{17} a^{14} - \frac{3}{17} a^{13} + \frac{6}{17} a^{12} + \frac{7}{17} a^{11} + \frac{8}{17} a^{10} - \frac{5}{17} a^{9} + \frac{2}{17} a^{8} - \frac{4}{17} a^{7} - \frac{5}{17} a^{6} - \frac{5}{17} a^{5} - \frac{1}{17} a^{4} - \frac{6}{17} a^{3} + \frac{2}{17} a^{2} + \frac{7}{17} a + \frac{4}{17}$

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:  \( 178967.823795 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.6652128125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
31Data not computed
1459Data not computed
121951Data not computed