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Properties

Label 20.0.92264062500...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}$
Root discriminant $17.71$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
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![1, -5, 10, -15, 30, -53, 50, -15, 45, -200, 308, -200, 45, -15, 50, -53, 30, -15, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 - 15*x^17 + 30*x^16 - 53*x^15 + 50*x^14 - 15*x^13 + 45*x^12 - 200*x^11 + 308*x^10 - 200*x^9 + 45*x^8 - 15*x^7 + 50*x^6 - 53*x^5 + 30*x^4 - 15*x^3 + 10*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 - 15*x^17 + 30*x^16 - 53*x^15 + 50*x^14 - 15*x^13 + 45*x^12 - 200*x^11 + 308*x^10 - 200*x^9 + 45*x^8 - 15*x^7 + 50*x^6 - 53*x^5 + 30*x^4 - 15*x^3 + 10*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} - 15 x^{17} + 30 x^{16} - 53 x^{15} + 50 x^{14} - 15 x^{13} + 45 x^{12} - 200 x^{11} + 308 x^{10} - 200 x^{9} + 45 x^{8} - 15 x^{7} + 50 x^{6} - 53 x^{5} + 30 x^{4} - 15 x^{3} + 10 x^{2} - 5 x + 1 \)

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:  \(9226406250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.71$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{482} a^{16} - \frac{2}{241} a^{15} - \frac{113}{241} a^{14} - \frac{35}{482} a^{13} - \frac{58}{241} a^{12} - \frac{112}{241} a^{11} + \frac{89}{241} a^{10} - \frac{3}{241} a^{9} - \frac{77}{241} a^{8} - \frac{3}{241} a^{7} + \frac{89}{241} a^{6} - \frac{112}{241} a^{5} + \frac{125}{482} a^{4} + \frac{103}{241} a^{3} - \frac{113}{241} a^{2} + \frac{237}{482} a - \frac{120}{241}$, $\frac{1}{482} a^{17} - \frac{1}{482} a^{15} + \frac{25}{482} a^{14} + \frac{113}{241} a^{13} + \frac{35}{482} a^{12} - \frac{118}{241} a^{11} + \frac{112}{241} a^{10} - \frac{89}{241} a^{9} - \frac{70}{241} a^{8} + \frac{77}{241} a^{7} + \frac{3}{241} a^{6} + \frac{193}{482} a^{5} + \frac{112}{241} a^{4} - \frac{125}{482} a^{3} - \frac{185}{482} a^{2} + \frac{113}{241} a - \frac{237}{482}$, $\frac{1}{3374} a^{18} + \frac{1}{3374} a^{17} + \frac{1}{3374} a^{16} + \frac{739}{3374} a^{15} - \frac{1165}{3374} a^{14} + \frac{191}{3374} a^{13} + \frac{386}{1687} a^{12} - \frac{230}{1687} a^{11} - \frac{522}{1687} a^{10} + \frac{799}{1687} a^{9} + \frac{94}{1687} a^{8} - \frac{167}{1687} a^{7} + \frac{555}{3374} a^{6} + \frac{933}{3374} a^{5} + \frac{1313}{3374} a^{4} - \frac{1585}{3374} a^{3} + \frac{71}{3374} a^{2} + \frac{463}{3374} a - \frac{479}{1687}$, $\frac{1}{114716} a^{19} - \frac{3}{57358} a^{18} + \frac{25}{57358} a^{17} - \frac{31}{114716} a^{16} + \frac{5093}{114716} a^{15} + \frac{28673}{57358} a^{14} + \frac{17711}{57358} a^{13} + \frac{25559}{114716} a^{12} + \frac{12976}{28679} a^{11} - \frac{10034}{28679} a^{10} - \frac{24}{119} a^{9} + \frac{11820}{28679} a^{8} - \frac{7523}{114716} a^{7} - \frac{8459}{28679} a^{6} + \frac{6168}{28679} a^{5} + \frac{34605}{114716} a^{4} - \frac{53479}{114716} a^{3} + \frac{13519}{28679} a^{2} - \frac{8510}{28679} a - \frac{3939}{16388}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{150735}{16388} a^{19} + \frac{334395}{8194} a^{18} - \frac{282660}{4097} a^{17} + \frac{1625535}{16388} a^{16} - \frac{3611967}{16388} a^{15} + \frac{1490395}{4097} a^{14} - \frac{1047240}{4097} a^{13} - \frac{71045}{16388} a^{12} - \frac{1715620}{4097} a^{11} + \frac{6574003}{4097} a^{10} - \frac{7901055}{4097} a^{9} + \frac{3104900}{4097} a^{8} + \frac{5965}{16388} a^{7} + \frac{606605}{4097} a^{6} - \frac{3086493}{8194} a^{5} + \frac{4526715}{16388} a^{4} - \frac{2012635}{16388} a^{3} + \frac{584645}{8194} a^{2} - \frac{439445}{8194} a + \frac{281439}{16388} \) (order $6$)
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:  \( 71455.53691328382 \)
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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 5.1.50000.1, 10.0.3037500000000.2, 10.0.607500000000.2, 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed