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Properties

Label 20.0.55612146387...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $24.39$
Ramified primes $5, 7, 17$
Class number $4$
Class group $[4]$
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![1019, -1485, 1594, -8077, 18576, -23909, 24538, -21832, 12919, -3573, 457, -323, 24, 7, 132, -50, -34, 14, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 9*x^18 + 14*x^17 - 34*x^16 - 50*x^15 + 132*x^14 + 7*x^13 + 24*x^12 - 323*x^11 + 457*x^10 - 3573*x^9 + 12919*x^8 - 21832*x^7 + 24538*x^6 - 23909*x^5 + 18576*x^4 - 8077*x^3 + 1594*x^2 - 1485*x + 1019)
 
gp: K = bnfinit(x^20 - 6*x^19 + 9*x^18 + 14*x^17 - 34*x^16 - 50*x^15 + 132*x^14 + 7*x^13 + 24*x^12 - 323*x^11 + 457*x^10 - 3573*x^9 + 12919*x^8 - 21832*x^7 + 24538*x^6 - 23909*x^5 + 18576*x^4 - 8077*x^3 + 1594*x^2 - 1485*x + 1019, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 9 x^{18} + 14 x^{17} - 34 x^{16} - 50 x^{15} + 132 x^{14} + 7 x^{13} + 24 x^{12} - 323 x^{11} + 457 x^{10} - 3573 x^{9} + 12919 x^{8} - 21832 x^{7} + 24538 x^{6} - 23909 x^{5} + 18576 x^{4} - 8077 x^{3} + 1594 x^{2} - 1485 x + 1019 \)

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:  \(5561214638787231316416015625=5^{10}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.39$
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:  $5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{15} + \frac{3}{7} a^{13} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{17423} a^{17} - \frac{99}{2489} a^{16} + \frac{2089}{17423} a^{15} + \frac{4917}{17423} a^{14} + \frac{5557}{17423} a^{13} - \frac{2691}{17423} a^{12} + \frac{6438}{17423} a^{11} + \frac{7078}{17423} a^{10} - \frac{2734}{17423} a^{9} - \frac{93}{17423} a^{8} - \frac{2004}{17423} a^{7} + \frac{8478}{17423} a^{6} - \frac{5176}{17423} a^{5} - \frac{6382}{17423} a^{4} + \frac{5590}{17423} a^{3} - \frac{5108}{17423} a^{2} - \frac{4367}{17423} a + \frac{4181}{17423}$, $\frac{1}{17423} a^{18} - \frac{272}{17423} a^{16} - \frac{1207}{2489} a^{15} - \frac{1870}{17423} a^{14} + \frac{2805}{17423} a^{13} + \frac{5836}{17423} a^{12} + \frac{478}{2489} a^{11} + \frac{1479}{17423} a^{10} + \frac{4352}{17423} a^{9} + \frac{8217}{17423} a^{8} + \frac{6079}{17423} a^{7} - \frac{1473}{17423} a^{6} + \frac{465}{2489} a^{5} + \frac{831}{2489} a^{4} + \frac{3345}{17423} a^{3} + \frac{125}{17423} a^{2} - \frac{7971}{17423} a - \frac{4741}{17423}$, $\frac{1}{234946247256159992305206132497} a^{19} + \frac{2342834798293126096696574}{234946247256159992305206132497} a^{18} + \frac{2773701612526569441331917}{234946247256159992305206132497} a^{17} + \frac{4787994441538939748091247856}{234946247256159992305206132497} a^{16} + \frac{78497725242472569355997607147}{234946247256159992305206132497} a^{15} - \frac{12255666703364960575903649443}{33563749608022856043600876071} a^{14} - \frac{8629572132155340835201265623}{33563749608022856043600876071} a^{13} + \frac{3461711438421196045027643907}{18072788250473845561938933269} a^{12} + \frac{55401306938677307593767220632}{234946247256159992305206132497} a^{11} + \frac{62249425974009754425039113218}{234946247256159992305206132497} a^{10} + \frac{93350891331110110613789035823}{234946247256159992305206132497} a^{9} + \frac{27649540352189114155565797955}{234946247256159992305206132497} a^{8} + \frac{99347486257448404143143747388}{234946247256159992305206132497} a^{7} + \frac{80622911871786079328425434947}{234946247256159992305206132497} a^{6} - \frac{103248347637044879776149436139}{234946247256159992305206132497} a^{5} + \frac{10765223624176443049371701034}{33563749608022856043600876071} a^{4} + \frac{12024606409774218849289861185}{33563749608022856043600876071} a^{3} - \frac{112716163489948336113100138168}{234946247256159992305206132497} a^{2} - \frac{66062735679700151848686161147}{234946247256159992305206132497} a - \frac{28720495231074640117705724632}{234946247256159992305206132497}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 69161.3662661 \)
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

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-595}) \), \(\Q(\sqrt{5}, \sqrt{-119})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.2.626668503125.1 x5, 10.0.74573551871875.1 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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$