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Properties

Label 20.0.16294596052...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 5^{17}\cdot 71^{10}$
Root discriminant $51.36$
Ramified primes $3, 5, 71$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 810, 3708, 8667, 11280, 8148, 25138, 31607, 37015, 16463, 6339, 6377, 1277, -994, -183, 2, 17, 20, -7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 7*x^18 + 20*x^17 + 17*x^16 + 2*x^15 - 183*x^14 - 994*x^13 + 1277*x^12 + 6377*x^11 + 6339*x^10 + 16463*x^9 + 37015*x^8 + 31607*x^7 + 25138*x^6 + 8148*x^5 + 11280*x^4 + 8667*x^3 + 3708*x^2 + 810*x + 81)
 
gp: K = bnfinit(x^20 - 3*x^19 - 7*x^18 + 20*x^17 + 17*x^16 + 2*x^15 - 183*x^14 - 994*x^13 + 1277*x^12 + 6377*x^11 + 6339*x^10 + 16463*x^9 + 37015*x^8 + 31607*x^7 + 25138*x^6 + 8148*x^5 + 11280*x^4 + 8667*x^3 + 3708*x^2 + 810*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 7 x^{18} + 20 x^{17} + 17 x^{16} + 2 x^{15} - 183 x^{14} - 994 x^{13} + 1277 x^{12} + 6377 x^{11} + 6339 x^{10} + 16463 x^{9} + 37015 x^{8} + 31607 x^{7} + 25138 x^{6} + 8148 x^{5} + 11280 x^{4} + 8667 x^{3} + 3708 x^{2} + 810 x + 81 \)

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:  \(16294596052685417602356719970703125=3^{8}\cdot 5^{17}\cdot 71^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.36$
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:  $3, 5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{2}{9} a^{12} + \frac{2}{9} a^{11} - \frac{1}{9} a^{10} - \frac{4}{9} a^{9} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{1}{3} a^{12} - \frac{2}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{2}{27} a^{14} + \frac{4}{27} a^{13} - \frac{4}{9} a^{12} - \frac{1}{3} a^{11} - \frac{7}{27} a^{10} + \frac{7}{27} a^{9} - \frac{2}{9} a^{8} + \frac{4}{9} a^{7} - \frac{7}{27} a^{6} - \frac{4}{9} a^{5} + \frac{2}{27} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{18} + \frac{1}{81} a^{17} + \frac{2}{81} a^{15} + \frac{13}{81} a^{14} - \frac{4}{27} a^{13} + \frac{4}{9} a^{12} - \frac{34}{81} a^{11} - \frac{29}{81} a^{10} - \frac{8}{27} a^{9} + \frac{13}{27} a^{8} - \frac{25}{81} a^{7} - \frac{4}{27} a^{6} + \frac{20}{81} a^{5} + \frac{10}{27} a^{4} - \frac{5}{27} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{201015766131707833712373774004826714937} a^{19} - \frac{346641561303528815081867824805089441}{67005255377235944570791258001608904979} a^{18} - \frac{3251374041559477696678067817894633187}{201015766131707833712373774004826714937} a^{17} - \frac{1815742957050534613213687768568559874}{201015766131707833712373774004826714937} a^{16} - \frac{4193815935987417191323624341050636365}{201015766131707833712373774004826714937} a^{15} - \frac{30710625440608130096127091835048867224}{201015766131707833712373774004826714937} a^{14} - \frac{3657111400844656336387879263546954007}{22335085125745314856930419333869634993} a^{13} + \frac{98360732910289202580293292721026882413}{201015766131707833712373774004826714937} a^{12} - \frac{5506974868218726271249236554965437265}{201015766131707833712373774004826714937} a^{11} + \frac{59425352542699057989899544093068410205}{201015766131707833712373774004826714937} a^{10} - \frac{32208180443735615815882508328444276220}{67005255377235944570791258001608904979} a^{9} - \frac{1439660769088635890893041998415729814}{201015766131707833712373774004826714937} a^{8} - \frac{73854198988296953054115995279580111509}{201015766131707833712373774004826714937} a^{7} + \frac{16003865110771105302204317969168036210}{201015766131707833712373774004826714937} a^{6} - \frac{89304412791619458702093621232222164194}{201015766131707833712373774004826714937} a^{5} + \frac{204483216918163220952791297099904154}{1810952848033403907318682648692132567} a^{4} + \frac{30984817953400114158764117020436564771}{67005255377235944570791258001608904979} a^{3} - \frac{2272391559558628073088460730035030214}{22335085125745314856930419333869634993} a^{2} - \frac{1361950013558321657828140067316123058}{7445028375248438285643473111289878331} a + \frac{6438905176740872264772153853268790}{2481676125082812761881157703763292777}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 1391760570.32 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-71}) \), 4.0.25205.1, 10.0.57086944308984375.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.10.13.7$x^{10} + 5 x^{4} + 10$$10$$1$$13$$D_5$$[3/2]_{2}$
$71$71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$