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Properties

Label 20.20.1918429393...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 5^{15}\cdot 239^{8}$
Root discriminant $51.78$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times D_5$ (as 20T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-101429, 185386, 616317, -651709, -1261123, 1017051, 1266895, -898678, -710001, 483926, 230822, -162403, -42360, 33772, 3737, -4198, -21, 284, -20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429)
 
gp: K = bnfinit(x^20 - 8*x^19 - 20*x^18 + 284*x^17 - 21*x^16 - 4198*x^15 + 3737*x^14 + 33772*x^13 - 42360*x^12 - 162403*x^11 + 230822*x^10 + 483926*x^9 - 710001*x^8 - 898678*x^7 + 1266895*x^6 + 1017051*x^5 - 1261123*x^4 - 651709*x^3 + 616317*x^2 + 185386*x - 101429, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 20 x^{18} + 284 x^{17} - 21 x^{16} - 4198 x^{15} + 3737 x^{14} + 33772 x^{13} - 42360 x^{12} - 162403 x^{11} + 230822 x^{10} + 483926 x^{9} - 710001 x^{8} - 898678 x^{7} + 1266895 x^{6} + 1017051 x^{5} - 1261123 x^{4} - 651709 x^{3} + 616317 x^{2} + 185386 x - 101429 \)

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:  \(19184293930982457734868438720703125=3^{10}\cdot 5^{15}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.78$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{239} a^{17} - \frac{22}{239} a^{16} - \frac{74}{239} a^{15} - \frac{68}{239} a^{14} - \frac{40}{239} a^{13} + \frac{89}{239} a^{12} - \frac{41}{239} a^{11} + \frac{22}{239} a^{10} + \frac{7}{239} a^{9} + \frac{18}{239} a^{8} + \frac{65}{239} a^{7} - \frac{44}{239} a^{6} + \frac{18}{239} a^{5} + \frac{62}{239} a^{3} + \frac{114}{239} a^{2} - \frac{60}{239} a - \frac{3}{239}$, $\frac{1}{207691} a^{18} - \frac{94}{207691} a^{17} - \frac{7333}{207691} a^{16} + \frac{26531}{207691} a^{15} - \frac{558}{18881} a^{14} - \frac{64668}{207691} a^{13} - \frac{87231}{207691} a^{12} - \frac{95972}{207691} a^{11} + \frac{27581}{207691} a^{10} - \frac{77444}{207691} a^{9} + \frac{31990}{207691} a^{8} + \frac{37340}{207691} a^{7} - \frac{51545}{207691} a^{6} - \frac{55310}{207691} a^{5} - \frac{18819}{207691} a^{4} + \frac{4428}{18881} a^{3} - \frac{2077}{18881} a^{2} - \frac{16954}{207691} a - \frac{90604}{207691}$, $\frac{1}{11938295449426789181460359} a^{19} - \frac{17480162307417229660}{11938295449426789181460359} a^{18} - \frac{567300216934378706788}{1085299586311526289223669} a^{17} + \frac{2335213626053064882426783}{11938295449426789181460359} a^{16} + \frac{4397819700582035146609}{11938295449426789181460359} a^{15} + \frac{502292987510885878016920}{11938295449426789181460359} a^{14} - \frac{78885982053690226459589}{628331339443515220076861} a^{13} - \frac{5148985795292811211684884}{11938295449426789181460359} a^{12} - \frac{335597785034784414088864}{11938295449426789181460359} a^{11} + \frac{2675237976696079139822791}{11938295449426789181460359} a^{10} + \frac{2034776873893554556464086}{11938295449426789181460359} a^{9} - \frac{120707799674259138763521}{1085299586311526289223669} a^{8} + \frac{5401293377531710291337127}{11938295449426789181460359} a^{7} - \frac{4663425337138317591779096}{11938295449426789181460359} a^{6} - \frac{5140390574695104237554389}{11938295449426789181460359} a^{5} + \frac{969503779882933584326823}{11938295449426789181460359} a^{4} + \frac{415419785483966756145447}{1085299586311526289223669} a^{3} - \frac{2629740126632412671936105}{11938295449426789181460359} a^{2} - \frac{3547195915827483707611086}{11938295449426789181460359} a + \frac{3775808185099302298091}{11938295449426789181460359}$

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:  \( 62134578326.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_5$ (as 20T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.12852225.1, 10.10.825898437253125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 $20$ R R $20$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
239Data not computed