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Properties

Label 20.0.44136757656...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $27.05$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![151, 400, -35, -570, -765, -86, 2880, 3050, 7205, 2670, 7257, 630, 4035, 10, 1340, -4, 255, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 25*x^18 + 255*x^16 - 4*x^15 + 1340*x^14 + 10*x^13 + 4035*x^12 + 630*x^11 + 7257*x^10 + 2670*x^9 + 7205*x^8 + 3050*x^7 + 2880*x^6 - 86*x^5 - 765*x^4 - 570*x^3 - 35*x^2 + 400*x + 151)
 
gp: K = bnfinit(x^20 + 25*x^18 + 255*x^16 - 4*x^15 + 1340*x^14 + 10*x^13 + 4035*x^12 + 630*x^11 + 7257*x^10 + 2670*x^9 + 7205*x^8 + 3050*x^7 + 2880*x^6 - 86*x^5 - 765*x^4 - 570*x^3 - 35*x^2 + 400*x + 151, 1)
 

Normalized defining polynomial

\( x^{20} + 25 x^{18} + 255 x^{16} - 4 x^{15} + 1340 x^{14} + 10 x^{13} + 4035 x^{12} + 630 x^{11} + 7257 x^{10} + 2670 x^{9} + 7205 x^{8} + 3050 x^{7} + 2880 x^{6} - 86 x^{5} - 765 x^{4} - 570 x^{3} - 35 x^{2} + 400 x + 151 \)

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:  \(44136757656250000000000000000=2^{16}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.05$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{7} a^{16} - \frac{1}{14} a^{15} - \frac{1}{7} a^{14} - \frac{1}{14} a^{13} + \frac{3}{14} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{6} + \frac{1}{14} a^{5} + \frac{2}{7} a^{4} + \frac{3}{14} a^{3} + \frac{1}{14} a^{2} + \frac{3}{14} a - \frac{3}{7}$, $\frac{1}{88541572993454126385186654946002} a^{19} - \frac{371122832267267147788941410993}{88541572993454126385186654946002} a^{18} + \frac{19039552894887310317651089565239}{88541572993454126385186654946002} a^{17} + \frac{9313808975470457687675140814092}{44270786496727063192593327473001} a^{16} + \frac{130421559412231288766822219302}{6324398070961009027513332496143} a^{15} + \frac{7445670584172724360243596288428}{44270786496727063192593327473001} a^{14} + \frac{2057278439416064971729597080118}{14756928832242354397531109157667} a^{13} - \frac{1063341400118521018254211536001}{6324398070961009027513332496143} a^{12} - \frac{3812776451804422714586690468860}{44270786496727063192593327473001} a^{11} + \frac{16727882592991343698092917660029}{88541572993454126385186654946002} a^{10} - \frac{11188176749924592453161298349769}{88541572993454126385186654946002} a^{9} - \frac{9021619482199514157041971243961}{88541572993454126385186654946002} a^{8} + \frac{5243029864451935081543566359029}{29513857664484708795062218315334} a^{7} + \frac{11101730838249973595002635192388}{44270786496727063192593327473001} a^{6} - \frac{1746826865786476316009822478998}{44270786496727063192593327473001} a^{5} - \frac{12107722187574907258193453152805}{29513857664484708795062218315334} a^{4} - \frac{4135574156460865623372838455599}{14756928832242354397531109157667} a^{3} - \frac{14264578918595036259161055677571}{29513857664484708795062218315334} a^{2} - \frac{7594831563594002425630194562739}{88541572993454126385186654946002} a - \frac{24019031509768088811390247305139}{88541572993454126385186654946002}$

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:  $9$
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:  \( 1767759.5386659107 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 5.1.50000.1, 10.0.42017500000000.4, 10.0.210087500000000.3, 10.2.12500000000.1

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 10 siblings: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$