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Properties

Label 20.0.10141885549...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 7^{16}$
Root discriminant $15.86$
Ramified primes $5, 7$
Class number $1$
Class group Trivial
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -100, 215, -370, 546, -690, 773, -775, 703, -635, 561, -445, 355, -275, 181, -110, 63, -30, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 30*x^17 + 63*x^16 - 110*x^15 + 181*x^14 - 275*x^13 + 355*x^12 - 445*x^11 + 561*x^10 - 635*x^9 + 703*x^8 - 775*x^7 + 773*x^6 - 690*x^5 + 546*x^4 - 370*x^3 + 215*x^2 - 100*x + 25)
 
gp: K = bnfinit(x^20 - 5*x^19 + 13*x^18 - 30*x^17 + 63*x^16 - 110*x^15 + 181*x^14 - 275*x^13 + 355*x^12 - 445*x^11 + 561*x^10 - 635*x^9 + 703*x^8 - 775*x^7 + 773*x^6 - 690*x^5 + 546*x^4 - 370*x^3 + 215*x^2 - 100*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 13 x^{18} - 30 x^{17} + 63 x^{16} - 110 x^{15} + 181 x^{14} - 275 x^{13} + 355 x^{12} - 445 x^{11} + 561 x^{10} - 635 x^{9} + 703 x^{8} - 775 x^{7} + 773 x^{6} - 690 x^{5} + 546 x^{4} - 370 x^{3} + 215 x^{2} - 100 x + 25 \)

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:  \(1014188554980499267578125=5^{15}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.86$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{3}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{15} - \frac{2}{5} a^{5} - \frac{1}{2}$, $\frac{1}{10} a^{16} - \frac{2}{5} a^{6} - \frac{1}{2} a$, $\frac{1}{50} a^{17} - \frac{1}{25} a^{16} + \frac{1}{50} a^{15} - \frac{1}{25} a^{14} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{3}{25} a^{7} - \frac{6}{25} a^{6} + \frac{3}{25} a^{5} - \frac{6}{25} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2}$, $\frac{1}{6950} a^{18} - \frac{19}{3475} a^{17} + \frac{104}{3475} a^{16} + \frac{127}{6950} a^{15} - \frac{193}{6950} a^{14} - \frac{43}{1390} a^{13} + \frac{21}{1390} a^{12} + \frac{31}{1390} a^{11} - \frac{117}{1390} a^{10} + \frac{133}{1390} a^{9} + \frac{1821}{6950} a^{8} + \frac{927}{6950} a^{7} - \frac{2597}{6950} a^{6} + \frac{2007}{6950} a^{5} + \frac{397}{6950} a^{4} - \frac{194}{695} a^{3} - \frac{59}{1390} a^{2} - \frac{13}{278} a + \frac{59}{139}$, $\frac{1}{1257950} a^{19} + \frac{11}{628975} a^{18} - \frac{5547}{1257950} a^{17} - \frac{16583}{1257950} a^{16} - \frac{8449}{628975} a^{15} + \frac{9039}{251590} a^{14} - \frac{6451}{251590} a^{13} + \frac{14913}{251590} a^{12} - \frac{3261}{251590} a^{11} - \frac{2161}{251590} a^{10} + \frac{164041}{1257950} a^{9} + \frac{461857}{1257950} a^{8} + \frac{132253}{1257950} a^{7} - \frac{96823}{1257950} a^{6} + \frac{361287}{1257950} a^{5} - \frac{5596}{125795} a^{4} - \frac{14231}{50318} a^{3} + \frac{44554}{125795} a^{2} - \frac{12424}{25159} a - \frac{12241}{50318}$

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:  \( -\frac{859753}{1257950} a^{19} + \frac{702303}{251590} a^{18} - \frac{7945187}{1257950} a^{17} + \frac{18442607}{1257950} a^{16} - \frac{37080189}{1257950} a^{15} + \frac{60154067}{1257950} a^{14} - \frac{19912729}{251590} a^{13} + \frac{28710731}{251590} a^{12} - \frac{34190167}{251590} a^{11} + \frac{44370669}{251590} a^{10} - \frac{273689903}{1257950} a^{9} + \frac{57710073}{251590} a^{8} - \frac{331948577}{1257950} a^{7} + \frac{353173137}{1257950} a^{6} - \frac{331564159}{1257950} a^{5} + \frac{139498466}{628975} a^{4} - \frac{20302194}{125795} a^{3} + \frac{12375067}{125795} a^{2} - \frac{1303214}{25159} a + \frac{454843}{25159} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 53376.7417145 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.300125.1 x5, 10.2.450375078125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.300125.1
Degree 10 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed