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Properties

Label 20.0.33630250781...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{16}\cdot 5^{23}$
Root discriminant $26.69$
Ramified primes $2, 3, 5$
Class number $5$ (GRH)
Class group $[5]$ (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![2401, -1715, -2205, 595, 3460, -7753, 1565, 4905, 2780, -6185, 5169, -2555, 590, -135, 95, -97, 70, -35, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 35*x^17 + 70*x^16 - 97*x^15 + 95*x^14 - 135*x^13 + 590*x^12 - 2555*x^11 + 5169*x^10 - 6185*x^9 + 2780*x^8 + 4905*x^7 + 1565*x^6 - 7753*x^5 + 3460*x^4 + 595*x^3 - 2205*x^2 - 1715*x + 2401)
 
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 35*x^17 + 70*x^16 - 97*x^15 + 95*x^14 - 135*x^13 + 590*x^12 - 2555*x^11 + 5169*x^10 - 6185*x^9 + 2780*x^8 + 4905*x^7 + 1565*x^6 - 7753*x^5 + 3460*x^4 + 595*x^3 - 2205*x^2 - 1715*x + 2401, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 15 x^{18} - 35 x^{17} + 70 x^{16} - 97 x^{15} + 95 x^{14} - 135 x^{13} + 590 x^{12} - 2555 x^{11} + 5169 x^{10} - 6185 x^{9} + 2780 x^{8} + 4905 x^{7} + 1565 x^{6} - 7753 x^{5} + 3460 x^{4} + 595 x^{3} - 2205 x^{2} - 1715 x + 2401 \)

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:  \(33630250781250000000000000000=2^{16}\cdot 3^{16}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.69$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{2}{25} a^{5} + \frac{12}{25} a^{3} + \frac{12}{25} a^{2} - \frac{1}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{7} + \frac{1}{25} a^{6} + \frac{2}{25} a^{5} + \frac{12}{25} a^{4} + \frac{12}{25} a^{2} + \frac{12}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{5} - \frac{6}{25}$, $\frac{1}{125} a^{11} + \frac{2}{125} a^{10} - \frac{1}{125} a^{6} - \frac{2}{125} a^{5} - \frac{31}{125} a - \frac{62}{125}$, $\frac{1}{125} a^{12} + \frac{1}{125} a^{10} - \frac{1}{125} a^{7} - \frac{1}{125} a^{5} - \frac{31}{125} a^{2} - \frac{31}{125}$, $\frac{1}{125} a^{13} - \frac{2}{125} a^{10} - \frac{1}{125} a^{8} + \frac{2}{125} a^{5} - \frac{31}{125} a^{3} + \frac{62}{125}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{10} - \frac{1}{125} a^{9} + \frac{1}{125} a^{5} - \frac{31}{125} a^{4} + \frac{31}{125}$, $\frac{1}{625} a^{15} + \frac{11}{625} a^{10} - \frac{43}{625} a^{5} + \frac{253}{625}$, $\frac{1}{625} a^{16} + \frac{1}{625} a^{11} + \frac{1}{125} a^{10} - \frac{33}{625} a^{6} - \frac{1}{125} a^{5} - \frac{62}{625} a - \frac{31}{125}$, $\frac{1}{13698125} a^{17} + \frac{7268}{13698125} a^{16} - \frac{487}{2739625} a^{15} + \frac{1402}{391375} a^{14} + \frac{1002}{391375} a^{13} - \frac{47564}{13698125} a^{12} + \frac{43348}{13698125} a^{11} + \frac{35211}{2739625} a^{10} - \frac{30304}{2739625} a^{9} - \frac{557}{391375} a^{8} - \frac{548468}{13698125} a^{7} + \frac{820151}{13698125} a^{6} + \frac{33113}{2739625} a^{5} + \frac{1319886}{2739625} a^{4} + \frac{574446}{2739625} a^{3} + \frac{1415578}{13698125} a^{2} - \frac{6723071}{13698125} a - \frac{119152}{391375}$, $\frac{1}{95886875} a^{18} + \frac{2}{95886875} a^{17} + \frac{30164}{95886875} a^{16} - \frac{4701}{13698125} a^{15} - \frac{3918}{2739625} a^{14} + \frac{247696}{95886875} a^{13} + \frac{121867}{95886875} a^{12} + \frac{178694}{95886875} a^{11} + \frac{563733}{95886875} a^{10} + \frac{44803}{2739625} a^{9} - \frac{1139128}{95886875} a^{8} - \frac{2576606}{95886875} a^{7} - \frac{10432}{3093125} a^{6} - \frac{5689084}{95886875} a^{5} + \frac{8898671}{19177375} a^{4} + \frac{28775218}{95886875} a^{3} + \frac{10649711}{95886875} a^{2} - \frac{2430964}{13698125} a - \frac{612344}{1956875}$, $\frac{1}{3356040625} a^{19} - \frac{12}{3356040625} a^{18} + \frac{99}{3356040625} a^{17} + \frac{286546}{479434375} a^{16} + \frac{2353}{4746875} a^{15} - \frac{459864}{3356040625} a^{14} + \frac{11819868}{3356040625} a^{13} - \frac{1945561}{3356040625} a^{12} - \frac{6977633}{3356040625} a^{11} + \frac{3441183}{479434375} a^{10} + \frac{740797}{108259375} a^{9} + \frac{52003766}{3356040625} a^{8} - \frac{202616357}{3356040625} a^{7} - \frac{181355421}{3356040625} a^{6} + \frac{35005072}{3356040625} a^{5} + \frac{494617378}{3356040625} a^{4} - \frac{97170911}{3356040625} a^{3} - \frac{62252954}{479434375} a^{2} + \frac{30119059}{68490625} a - \frac{2302909}{9784375}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( \frac{22538}{95886875} a^{19} - \frac{101421}{95886875} a^{18} + \frac{300177}{95886875} a^{17} - \frac{22538}{3093125} a^{16} + \frac{202842}{13698125} a^{15} - \frac{1949537}{95886875} a^{14} + \frac{2152379}{95886875} a^{13} - \frac{485239}{13698125} a^{12} + \frac{428222}{3093125} a^{11} - \frac{53065721}{95886875} a^{10} + \frac{97014821}{95886875} a^{9} - \frac{121457282}{95886875} a^{8} + \frac{66644384}{95886875} a^{7} + \frac{44298439}{95886875} a^{6} + \frac{134405363}{95886875} a^{5} - \frac{79716906}{95886875} a^{4} + \frac{2186186}{13698125} a^{3} + \frac{121005001}{95886875} a^{2} + \frac{236649}{1956875} a - \frac{1104362}{1956875} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1373081.37212 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.4050000.4 x5, 10.2.82012500000000.5 x5

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

Sibling fields

Degree 5 sibling: 5.1.4050000.4
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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
3Data not computed
5Data not computed