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Properties

Label 20.0.59385313411...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 239^{10}$
Root discriminant $34.57$
Ramified primes $5, 239$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -817, 3436, -7417, 17988, -30332, 51585, -66389, 79306, -84235, 78193, -59462, 36392, -18569, 8069, -2998, 937, -244, 52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 52*x^18 - 244*x^17 + 937*x^16 - 2998*x^15 + 8069*x^14 - 18569*x^13 + 36392*x^12 - 59462*x^11 + 78193*x^10 - 84235*x^9 + 79306*x^8 - 66389*x^7 + 51585*x^6 - 30332*x^5 + 17988*x^4 - 7417*x^3 + 3436*x^2 - 817*x + 361)
 
gp: K = bnfinit(x^20 - 8*x^19 + 52*x^18 - 244*x^17 + 937*x^16 - 2998*x^15 + 8069*x^14 - 18569*x^13 + 36392*x^12 - 59462*x^11 + 78193*x^10 - 84235*x^9 + 79306*x^8 - 66389*x^7 + 51585*x^6 - 30332*x^5 + 17988*x^4 - 7417*x^3 + 3436*x^2 - 817*x + 361, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 52 x^{18} - 244 x^{17} + 937 x^{16} - 2998 x^{15} + 8069 x^{14} - 18569 x^{13} + 36392 x^{12} - 59462 x^{11} + 78193 x^{10} - 84235 x^{9} + 79306 x^{8} - 66389 x^{7} + 51585 x^{6} - 30332 x^{5} + 17988 x^{4} - 7417 x^{3} + 3436 x^{2} - 817 x + 361 \)

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:  \(5938531341125635825289072265625=5^{10}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.57$
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:  $5, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{2}$, $\frac{1}{161} a^{17} - \frac{10}{161} a^{16} - \frac{9}{161} a^{15} - \frac{8}{161} a^{14} + \frac{5}{23} a^{13} + \frac{10}{161} a^{12} - \frac{78}{161} a^{11} - \frac{15}{161} a^{10} - \frac{7}{23} a^{9} + \frac{8}{23} a^{8} - \frac{12}{161} a^{7} - \frac{8}{161} a^{6} - \frac{31}{161} a^{5} - \frac{40}{161} a^{4} - \frac{60}{161} a^{3} - \frac{41}{161} a^{2} + \frac{58}{161} a + \frac{51}{161}$, $\frac{1}{68747} a^{18} + \frac{19}{68747} a^{17} + \frac{73}{2989} a^{16} + \frac{254}{9821} a^{15} + \frac{838}{68747} a^{14} + \frac{1073}{9821} a^{13} - \frac{19177}{68747} a^{12} - \frac{606}{2989} a^{11} - \frac{24749}{68747} a^{10} - \frac{8}{68747} a^{9} - \frac{1222}{9821} a^{8} - \frac{12822}{68747} a^{7} - \frac{30554}{68747} a^{6} + \frac{2271}{9821} a^{5} + \frac{16950}{68747} a^{4} + \frac{31776}{68747} a^{3} + \frac{14187}{68747} a^{2} - \frac{34124}{68747} a - \frac{10343}{68747}$, $\frac{1}{4055949116173286095073822147013581} a^{19} - \frac{18485887718179559677406873674}{4055949116173286095073822147013581} a^{18} - \frac{11799988188818883422507958679540}{4055949116173286095073822147013581} a^{17} + \frac{4090581563829357436053358858117}{176345613746664612829296615087547} a^{16} - \frac{5429852842620975413577926098820}{176345613746664612829296615087547} a^{15} + \frac{216002068311522293941642681229861}{4055949116173286095073822147013581} a^{14} + \frac{1907631683863629254442566796629316}{4055949116173286095073822147013581} a^{13} + \frac{282634369733818866901117998513633}{4055949116173286095073822147013581} a^{12} + \frac{1219741205590042175695056368121826}{4055949116173286095073822147013581} a^{11} - \frac{321332004570907568381778088086720}{4055949116173286095073822147013581} a^{10} - \frac{667261910131395412098613739517732}{4055949116173286095073822147013581} a^{9} + \frac{227644479769277742926884509602063}{4055949116173286095073822147013581} a^{8} - \frac{104840800726295261167995965267570}{213471006114383478688095902474399} a^{7} + \frac{1734037508281952484863638206710369}{4055949116173286095073822147013581} a^{6} - \frac{96699398124268563074709846781485}{213471006114383478688095902474399} a^{5} - \frac{989884321000473063958134001804381}{4055949116173286095073822147013581} a^{4} + \frac{1484579125448610750404803239046005}{4055949116173286095073822147013581} a^{3} - \frac{115401595132587598386004931648095}{579421302310469442153403163859083} a^{2} - \frac{8589090410074720395966260122976}{25192230535237801832756659298221} a - \frac{53932343907355367471826213467285}{213471006114383478688095902474399}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 892472.509302 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-239}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1195}) \), \(\Q(\sqrt{5}, \sqrt{-239})\), 5.1.57121.1 x5, 10.0.779811265199.1, 10.2.10196277003125.1 x5, 10.0.2436910203746875.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
239Data not computed