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Properties

Label 20.0.13256483588...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 23^{10}$
Root discriminant $32.07$
Ramified primes $2, 5, 23$
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![48220, -17700, -36870, -55690, 49621, 77300, -32170, -43440, 1091, 15670, 5160, -3590, -1584, 20, 290, 80, 11, -10, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 10*x^17 + 11*x^16 + 80*x^15 + 290*x^14 + 20*x^13 - 1584*x^12 - 3590*x^11 + 5160*x^10 + 15670*x^9 + 1091*x^8 - 43440*x^7 - 32170*x^6 + 77300*x^5 + 49621*x^4 - 55690*x^3 - 36870*x^2 - 17700*x + 48220)
 
gp: K = bnfinit(x^20 - 10*x^18 - 10*x^17 + 11*x^16 + 80*x^15 + 290*x^14 + 20*x^13 - 1584*x^12 - 3590*x^11 + 5160*x^10 + 15670*x^9 + 1091*x^8 - 43440*x^7 - 32170*x^6 + 77300*x^5 + 49621*x^4 - 55690*x^3 - 36870*x^2 - 17700*x + 48220, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 10 x^{17} + 11 x^{16} + 80 x^{15} + 290 x^{14} + 20 x^{13} - 1584 x^{12} - 3590 x^{11} + 5160 x^{10} + 15670 x^{9} + 1091 x^{8} - 43440 x^{7} - 32170 x^{6} + 77300 x^{5} + 49621 x^{4} - 55690 x^{3} - 36870 x^{2} - 17700 x + 48220 \)

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:  \(1325648358836768000000000000000=2^{20}\cdot 5^{15}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.07$
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:  $2, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{4} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{5}{12} a^{6} + \frac{5}{12} a^{5} - \frac{1}{4} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{120} a^{16} - \frac{1}{30} a^{15} - \frac{1}{60} a^{14} + \frac{11}{60} a^{13} - \frac{1}{15} a^{12} - \frac{1}{10} a^{11} - \frac{2}{15} a^{10} + \frac{1}{20} a^{9} + \frac{1}{15} a^{8} - \frac{9}{20} a^{7} - \frac{1}{10} a^{6} + \frac{13}{30} a^{5} - \frac{1}{120} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{120} a^{17} + \frac{1}{60} a^{15} - \frac{1}{20} a^{14} + \frac{1}{6} a^{13} - \frac{1}{30} a^{12} + \frac{2}{15} a^{11} + \frac{11}{60} a^{10} - \frac{1}{15} a^{9} + \frac{3}{20} a^{8} + \frac{4}{15} a^{7} - \frac{3}{10} a^{6} + \frac{7}{120} a^{5} + \frac{1}{20} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{28512480} a^{18} - \frac{9625}{2851248} a^{17} - \frac{39313}{9504160} a^{16} - \frac{309341}{14256240} a^{15} + \frac{246251}{14256240} a^{14} - \frac{42013}{14256240} a^{13} - \frac{784027}{3564060} a^{12} - \frac{3406853}{14256240} a^{11} + \frac{1657867}{7128120} a^{10} + \frac{1665073}{7128120} a^{9} + \frac{1474121}{7128120} a^{8} + \frac{2206543}{4752080} a^{7} - \frac{1422727}{9504160} a^{6} + \frac{375782}{891015} a^{5} + \frac{3936551}{28512480} a^{4} - \frac{414567}{950416} a^{3} - \frac{255279}{950416} a^{2} - \frac{57149}{475208} a - \frac{192851}{475208}$, $\frac{1}{34727520439428844645114993618722240} a^{19} - \frac{553386786927159929284672699}{34727520439428844645114993618722240} a^{18} + \frac{19766140690223540659063024484141}{11575840146476281548371664539574080} a^{17} + \frac{666613961057986982692744279307}{11575840146476281548371664539574080} a^{16} + \frac{4941624901412570949313186749678}{180872502288691899193307258430845} a^{15} + \frac{25226302145035741907648623098115}{434094005492860558063937420234028} a^{14} - \frac{1052838490015396586835150694647359}{17363760219714422322557496809361120} a^{13} - \frac{958844628157708426282434607083051}{5787920073238140774185832269787040} a^{12} + \frac{862579300757336488078758826265047}{3472752043942884464511499361872224} a^{11} - \frac{267108895139215418823166539245957}{1446980018309535193546458067446760} a^{10} + \frac{124880051089295290887200557176968}{542617506866075697579921775292535} a^{9} - \frac{1142223837988010052102889432586093}{17363760219714422322557496809361120} a^{8} - \frac{5602732171056822883146530626355269}{11575840146476281548371664539574080} a^{7} - \frac{2262653943363578500650477991167803}{34727520439428844645114993618722240} a^{6} - \frac{5390180115306517560531752600820899}{11575840146476281548371664539574080} a^{5} + \frac{16178176149834939470315965237503791}{34727520439428844645114993618722240} a^{4} + \frac{61620518515948750038876462108031}{217047002746430279031968710117014} a^{3} - \frac{1648400214882243367412085625399561}{3472752043942884464511499361872224} a^{2} - \frac{109077836913306950880805047840437}{868188010985721116127874840468056} a - \frac{756581635866431486176789919995343}{1736376021971442232255749680936112}$

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:  \( 8895897.14135 \) (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

\(\Q(\sqrt{5}) \), 4.0.1058000.1, 5.1.1058000.1 x5, 10.2.5596820000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1058000.1
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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$