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Properties

Label 20.0.28963243469...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 3469^{4}$
Root discriminant $29.72$
Ramified primes $2, 5, 3469$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $D_5^2:C_4$ (as 20T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 32, -72, 126, 59, 462, 218, 963, 110, 986, -306, 410, -96, 96, -52, 51, -17, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 17*x^17 + 51*x^16 - 52*x^15 + 96*x^14 - 96*x^13 + 410*x^12 - 306*x^11 + 986*x^10 + 110*x^9 + 963*x^8 + 218*x^7 + 462*x^6 + 59*x^5 + 126*x^4 - 72*x^3 + 32*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 17*x^17 + 51*x^16 - 52*x^15 + 96*x^14 - 96*x^13 + 410*x^12 - 306*x^11 + 986*x^10 + 110*x^9 + 963*x^8 + 218*x^7 + 462*x^6 + 59*x^5 + 126*x^4 - 72*x^3 + 32*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 17 x^{17} + 51 x^{16} - 52 x^{15} + 96 x^{14} - 96 x^{13} + 410 x^{12} - 306 x^{11} + 986 x^{10} + 110 x^{9} + 963 x^{8} + 218 x^{7} + 462 x^{6} + 59 x^{5} + 126 x^{4} - 72 x^{3} + 32 x^{2} - 7 x + 1 \)

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:  \(289632434699042000000000000000=2^{16}\cdot 5^{15}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.72$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3077966998573138564031821691} a^{19} + \frac{1062081806641096247619842126}{3077966998573138564031821691} a^{18} - \frac{152285087466157205475759029}{3077966998573138564031821691} a^{17} + \frac{239210546917667335394659156}{3077966998573138564031821691} a^{16} + \frac{62330953141383875453478198}{3077966998573138564031821691} a^{15} + \frac{1480680229861509450525247209}{3077966998573138564031821691} a^{14} + \frac{37951675226620570206297602}{3077966998573138564031821691} a^{13} + \frac{3010735759436489144978264}{3077966998573138564031821691} a^{12} - \frac{802538582238616692500262684}{3077966998573138564031821691} a^{11} - \frac{938473788862065763410084687}{3077966998573138564031821691} a^{10} - \frac{424038846131196280332428985}{3077966998573138564031821691} a^{9} + \frac{231497108479981342554351535}{3077966998573138564031821691} a^{8} - \frac{400760873154151549325508058}{3077966998573138564031821691} a^{7} - \frac{1109152252548824445174453803}{3077966998573138564031821691} a^{6} - \frac{1090267545151189241395768950}{3077966998573138564031821691} a^{5} + \frac{418069586552870374181337346}{3077966998573138564031821691} a^{4} - \frac{767803062558669659339296211}{3077966998573138564031821691} a^{3} - \frac{270401247012804339660738221}{3077966998573138564031821691} a^{2} - \frac{963414528995040636141768957}{3077966998573138564031821691} a + \frac{1037652713137098212867728308}{3077966998573138564031821691}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( -\frac{1203104565908224793236193664}{3077966998573138564031821691} a^{19} + \frac{4415513002367227037224273241}{3077966998573138564031821691} a^{18} - \frac{10473961110993488034462416992}{3077966998573138564031821691} a^{17} + \frac{16620014862007669988572045657}{3077966998573138564031821691} a^{16} - \frac{54970060289363437336914195732}{3077966998573138564031821691} a^{15} + \frac{42964884223645089243769385080}{3077966998573138564031821691} a^{14} - \frac{96605373365909500698398685100}{3077966998573138564031821691} a^{13} + \frac{79588266900424581638875535735}{3077966998573138564031821691} a^{12} - \frac{458562565486070814210595907642}{3077966998573138564031821691} a^{11} + \frac{209284282798069359953407037186}{3077966998573138564031821691} a^{10} - \frac{1077903559987066780705970943302}{3077966998573138564031821691} a^{9} - \frac{509369356809538310261880981513}{3077966998573138564031821691} a^{8} - \frac{1234808747659704088807174321796}{3077966998573138564031821691} a^{7} - \frac{636506717280422281161117372845}{3077966998573138564031821691} a^{6} - \frac{663555334280763985750921599392}{3077966998573138564031821691} a^{5} - \frac{252028067672038791272818241293}{3077966998573138564031821691} a^{4} - \frac{178586204309872031577806521510}{3077966998573138564031821691} a^{3} + \frac{44176255265660351161562230711}{3077966998573138564031821691} a^{2} - \frac{9255493036938140331681231228}{3077966998573138564031821691} a + \frac{1030455603024976423645868217}{3077966998573138564031821691} \) (order $10$)
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 20T93):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$5.4.3.2$x^{4} - 20$$4$$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