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Properties

Label 20.20.1438759916...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{6}\cdot 28162171^{2}$
Root discriminant $51.04$
Ramified primes $2, 5, 11, 28162171$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, -147741, 0, 533489, 0, -870342, 0, 734019, 0, -342567, 0, 92035, 0, -14363, 0, 1264, 0, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 57*x^18 + 1264*x^16 - 14363*x^14 + 92035*x^12 - 342567*x^10 + 734019*x^8 - 870342*x^6 + 533489*x^4 - 147741*x^2 + 14641)
 
gp: K = bnfinit(x^20 - 57*x^18 + 1264*x^16 - 14363*x^14 + 92035*x^12 - 342567*x^10 + 734019*x^8 - 870342*x^6 + 533489*x^4 - 147741*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{20} - 57 x^{18} + 1264 x^{16} - 14363 x^{14} + 92035 x^{12} - 342567 x^{10} + 734019 x^{8} - 870342 x^{6} + 533489 x^{4} - 147741 x^{2} + 14641 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14387599164522371678218240000000000=2^{20}\cdot 5^{10}\cdot 11^{6}\cdot 28162171^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.04$
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:  $2, 5, 11, 28162171$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{12} - \frac{1}{11} a^{10} + \frac{3}{11} a^{8} - \frac{2}{11} a^{6} - \frac{5}{11} a^{4}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{13} - \frac{1}{11} a^{11} + \frac{3}{11} a^{9} - \frac{2}{11} a^{7} - \frac{5}{11} a^{5}$, $\frac{1}{363} a^{16} - \frac{2}{363} a^{14} - \frac{59}{121} a^{12} - \frac{19}{363} a^{10} - \frac{41}{121} a^{8} + \frac{116}{363} a^{6} - \frac{1}{3} a^{4} - \frac{10}{33} a^{2} - \frac{1}{3}$, $\frac{1}{363} a^{17} - \frac{2}{363} a^{15} - \frac{59}{121} a^{13} - \frac{19}{363} a^{11} - \frac{41}{121} a^{9} + \frac{116}{363} a^{7} - \frac{1}{3} a^{5} - \frac{10}{33} a^{3} - \frac{1}{3} a$, $\frac{1}{3311180141874940599} a^{18} - \frac{1288611717271260}{1103726713958313533} a^{16} + \frac{129416800713478013}{3311180141874940599} a^{14} - \frac{194585990454636877}{3311180141874940599} a^{12} + \frac{346074913945066645}{3311180141874940599} a^{10} + \frac{896460610458343871}{3311180141874940599} a^{8} + \frac{19849116588118330}{100338792178028503} a^{6} - \frac{137825637070934042}{301016376534085509} a^{4} - \frac{7171416975499391}{27365125139462319} a^{2} - \frac{787387927494043}{2487738649042029}$, $\frac{1}{3311180141874940599} a^{19} - \frac{1288611717271260}{1103726713958313533} a^{17} + \frac{129416800713478013}{3311180141874940599} a^{15} - \frac{194585990454636877}{3311180141874940599} a^{13} + \frac{346074913945066645}{3311180141874940599} a^{11} + \frac{896460610458343871}{3311180141874940599} a^{9} + \frac{19849116588118330}{100338792178028503} a^{7} - \frac{137825637070934042}{301016376534085509} a^{5} - \frac{7171416975499391}{27365125139462319} a^{3} - \frac{787387927494043}{2487738649042029} a$

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:  $19$
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:  \( 45998273504.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.968074628125.1

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 32 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
28162171Data not computed