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Properties

Label 20.0.10047349159...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 3469^{5}$
Root discriminant $44.68$
Ramified primes $2, 5, 3469$
Class number $288$ (GRH)
Class group $[2, 4, 36]$ (GRH)
Galois group $D_5^2:C_4$ (as 20T94)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![99301, 51356, 212471, -56358, 172159, -178652, 145974, -119082, 78071, -44850, 30841, -13648, 6887, -3934, 1254, -570, 249, -50, 27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 27*x^18 - 50*x^17 + 249*x^16 - 570*x^15 + 1254*x^14 - 3934*x^13 + 6887*x^12 - 13648*x^11 + 30841*x^10 - 44850*x^9 + 78071*x^8 - 119082*x^7 + 145974*x^6 - 178652*x^5 + 172159*x^4 - 56358*x^3 + 212471*x^2 + 51356*x + 99301)
 
gp: K = bnfinit(x^20 - 2*x^19 + 27*x^18 - 50*x^17 + 249*x^16 - 570*x^15 + 1254*x^14 - 3934*x^13 + 6887*x^12 - 13648*x^11 + 30841*x^10 - 44850*x^9 + 78071*x^8 - 119082*x^7 + 145974*x^6 - 178652*x^5 + 172159*x^4 - 56358*x^3 + 212471*x^2 + 51356*x + 99301, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 27 x^{18} - 50 x^{17} + 249 x^{16} - 570 x^{15} + 1254 x^{14} - 3934 x^{13} + 6887 x^{12} - 13648 x^{11} + 30841 x^{10} - 44850 x^{9} + 78071 x^{8} - 119082 x^{7} + 145974 x^{6} - 178652 x^{5} + 172159 x^{4} - 56358 x^{3} + 212471 x^{2} + 51356 x + 99301 \)

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:  \(1004734915970976698000000000000000=2^{16}\cdot 5^{15}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.68$
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:  $2, 5, 3469$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{15} + \frac{1}{5} a^{13} - \frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{8} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{3}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{15} + \frac{1}{5} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{18} + \frac{1}{10} a^{15} + \frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{244032400950275062871096572126884993979318033951910} a^{19} - \frac{777755799596484253098988499637832088697061642011}{122016200475137531435548286063442496989659016975955} a^{18} - \frac{2452855089756135277127846513590755488970220304258}{122016200475137531435548286063442496989659016975955} a^{17} - \frac{2976013850851086688011894138685437065613298909541}{122016200475137531435548286063442496989659016975955} a^{16} - \frac{42527557711082119795678350971512760782153949098727}{244032400950275062871096572126884993979318033951910} a^{15} + \frac{3680604966848115247170700952331134394635036464858}{24403240095027506287109657212688499397931803395191} a^{14} - \frac{38031605426861737174745873717539013272760530135083}{244032400950275062871096572126884993979318033951910} a^{13} + \frac{1265192104301501616996204788051153616597175411293}{48806480190055012574219314425376998795863606790382} a^{12} - \frac{52449193152166101775454288988942795855864751804707}{244032400950275062871096572126884993979318033951910} a^{11} - \frac{11510437666856694726857148830258270682638504508628}{122016200475137531435548286063442496989659016975955} a^{10} - \frac{49003397416136736504872433610978067095630990221176}{122016200475137531435548286063442496989659016975955} a^{9} + \frac{55266847065501605995495346473005689757293607641}{728454928209776307077900215304134310386023981946} a^{8} + \frac{3411447563688232306479994228533075830963587547790}{24403240095027506287109657212688499397931803395191} a^{7} - \frac{9965379395963334952847000838623855893059892078725}{24403240095027506287109657212688499397931803395191} a^{6} - \frac{22609008381819713589377356015193463859687664432562}{122016200475137531435548286063442496989659016975955} a^{5} - \frac{3496362633083979589505603968973971342900684431679}{10610104389142394037873764005516738868666001476170} a^{4} + \frac{39559218307582611376513620457620233651561484482231}{122016200475137531435548286063442496989659016975955} a^{3} + \frac{49924024557654063363224626160447659931508111206514}{122016200475137531435548286063442496989659016975955} a^{2} + \frac{11806653230561684155119288613237648993450456067071}{244032400950275062871096572126884993979318033951910} a - \frac{55214739995747661063479166301103105890897056893843}{122016200475137531435548286063442496989659016975955}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{36}$, which has order $288$ (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:  \( 828338.933858 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T94):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.433625.1, 10.10.9627168800000.1

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 R $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{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}$
3469Data not computed