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Properties

Label 20.0.38944592868...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{15}\cdot 43^{10}$
Root discriminant $37.98$
Ramified primes $3, 5, 43$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
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![625, 0, -875, -375, 3025, -975, -1800, 2475, 6610, 1470, -575, -840, 286, -177, 102, -45, 31, -15, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 15*x^17 + 31*x^16 - 45*x^15 + 102*x^14 - 177*x^13 + 286*x^12 - 840*x^11 - 575*x^10 + 1470*x^9 + 6610*x^8 + 2475*x^7 - 1800*x^6 - 975*x^5 + 3025*x^4 - 375*x^3 - 875*x^2 + 625)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 15*x^17 + 31*x^16 - 45*x^15 + 102*x^14 - 177*x^13 + 286*x^12 - 840*x^11 - 575*x^10 + 1470*x^9 + 6610*x^8 + 2475*x^7 - 1800*x^6 - 975*x^5 + 3025*x^4 - 375*x^3 - 875*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 15 x^{17} + 31 x^{16} - 45 x^{15} + 102 x^{14} - 177 x^{13} + 286 x^{12} - 840 x^{11} - 575 x^{10} + 1470 x^{9} + 6610 x^{8} + 2475 x^{7} - 1800 x^{6} - 975 x^{5} + 3025 x^{4} - 375 x^{3} - 875 x^{2} + 625 \)

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:  \(38944592868564502539093017578125=3^{10}\cdot 5^{15}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.98$
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:  $3, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} - \frac{3}{10} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{3}{10} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{3}{10} a^{7} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a$, $\frac{1}{10} a^{14} - \frac{1}{2} a^{11} - \frac{2}{5} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{50} a^{15} - \frac{1}{50} a^{14} - \frac{1}{25} a^{13} - \frac{19}{50} a^{11} + \frac{1}{5} a^{10} + \frac{1}{25} a^{9} + \frac{23}{50} a^{8} - \frac{12}{25} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{12900} a^{16} + \frac{7}{4300} a^{15} + \frac{97}{4300} a^{14} - \frac{69}{2150} a^{13} + \frac{31}{2150} a^{12} + \frac{1399}{4300} a^{11} - \frac{583}{2150} a^{10} - \frac{99}{1075} a^{9} - \frac{131}{300} a^{8} + \frac{987}{2150} a^{7} + \frac{23}{430} a^{6} - \frac{293}{860} a^{5} - \frac{39}{86} a^{4} + \frac{76}{215} a^{3} - \frac{29}{172} a^{2} + \frac{19}{172} a + \frac{1}{516}$, $\frac{1}{12900} a^{17} + \frac{9}{1075} a^{15} - \frac{111}{4300} a^{14} + \frac{52}{1075} a^{13} + \frac{97}{4300} a^{12} - \frac{789}{4300} a^{11} + \frac{1}{430} a^{10} - \frac{809}{12900} a^{9} - \frac{47}{4300} a^{8} - \frac{287}{1075} a^{7} + \frac{289}{860} a^{6} + \frac{259}{860} a^{5} - \frac{139}{430} a^{4} + \frac{7}{860} a^{3} - \frac{15}{43} a^{2} - \frac{41}{129} a + \frac{79}{172}$, $\frac{1}{155509500} a^{18} + \frac{69}{25918250} a^{17} - \frac{793}{38877375} a^{16} - \frac{3645}{414692} a^{15} + \frac{452641}{25918250} a^{14} + \frac{282453}{10367300} a^{13} - \frac{727721}{51836500} a^{12} - \frac{5672631}{12959125} a^{11} + \frac{9441691}{155509500} a^{10} + \frac{3706661}{10367300} a^{9} - \frac{7405897}{15550950} a^{8} - \frac{3593511}{10367300} a^{7} - \frac{2115701}{10367300} a^{6} - \frac{154497}{518365} a^{5} + \frac{181739}{2073460} a^{4} + \frac{88162}{518365} a^{3} - \frac{339491}{1555095} a^{2} - \frac{198641}{414692} a + \frac{2621}{622038}$, $\frac{1}{7643057796272689500} a^{19} + \frac{11522862149}{7643057796272689500} a^{18} - \frac{117417614996357}{7643057796272689500} a^{17} + \frac{7165903711281}{509537186418179300} a^{16} - \frac{8546470590512323}{2547685932090896500} a^{15} - \frac{10812576312646807}{254768593209089650} a^{14} + \frac{38208427171628587}{1273842966045448250} a^{13} + \frac{16977363505034684}{636921483022724125} a^{12} + \frac{760051514441534333}{3821528898136344750} a^{11} + \frac{28991571007134977}{764305779627268950} a^{10} + \frac{4296690860210017}{76430577962726895} a^{9} + \frac{42144395205144971}{254768593209089650} a^{8} + \frac{77321744458609721}{254768593209089650} a^{7} - \frac{8484898348255694}{25476859320908965} a^{6} - \frac{1983466825021810}{5095371864181793} a^{5} - \frac{5733072654144749}{20381487456727172} a^{4} + \frac{1494882296488127}{61144462370181516} a^{3} - \frac{29893365894405389}{61144462370181516} a^{2} + \frac{22644967035878291}{61144462370181516} a - \frac{4811418416790699}{20381487456727172}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 26079539.7398 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 4.0.2080125.1, 5.1.2080125.1 x5, 10.2.21634600078125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.2080125.1
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$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}$
$43$43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$