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Properties

Label 20.0.10719236614...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 37^{8}$
Root discriminant $14.17$
Ramified primes $5, 37$
Class number $1$
Class group Trivial
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 25, 40, -70, -14, -85, 292, -325, 208, -140, 119, -70, 30, -25, 34, -35, 28, -20, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 12*x^18 - 20*x^17 + 28*x^16 - 35*x^15 + 34*x^14 - 25*x^13 + 30*x^12 - 70*x^11 + 119*x^10 - 140*x^9 + 208*x^8 - 325*x^7 + 292*x^6 - 85*x^5 - 14*x^4 - 70*x^3 + 40*x^2 + 25*x + 25)
 
gp: K = bnfinit(x^20 - 5*x^19 + 12*x^18 - 20*x^17 + 28*x^16 - 35*x^15 + 34*x^14 - 25*x^13 + 30*x^12 - 70*x^11 + 119*x^10 - 140*x^9 + 208*x^8 - 325*x^7 + 292*x^6 - 85*x^5 - 14*x^4 - 70*x^3 + 40*x^2 + 25*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 12 x^{18} - 20 x^{17} + 28 x^{16} - 35 x^{15} + 34 x^{14} - 25 x^{13} + 30 x^{12} - 70 x^{11} + 119 x^{10} - 140 x^{9} + 208 x^{8} - 325 x^{7} + 292 x^{6} - 85 x^{5} - 14 x^{4} - 70 x^{3} + 40 x^{2} + 25 x + 25 \)

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:  \(107192366147491455078125=5^{15}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.17$
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:  $5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{11} + \frac{2}{5} a^{9} - \frac{1}{2} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{15} + \frac{2}{5} a^{5} - \frac{1}{2}$, $\frac{1}{30} a^{16} - \frac{1}{30} a^{15} - \frac{1}{30} a^{14} - \frac{1}{30} a^{12} - \frac{1}{30} a^{11} + \frac{1}{15} a^{10} - \frac{11}{30} a^{9} - \frac{1}{10} a^{8} + \frac{2}{15} a^{7} + \frac{11}{30} a^{6} - \frac{1}{6} a^{5} + \frac{4}{15} a^{4} - \frac{1}{30} a^{3} - \frac{7}{30} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{30} a^{17} + \frac{1}{30} a^{15} - \frac{1}{30} a^{14} - \frac{1}{30} a^{13} + \frac{1}{30} a^{12} + \frac{1}{30} a^{11} - \frac{1}{10} a^{10} + \frac{1}{30} a^{9} + \frac{7}{30} a^{8} - \frac{1}{2} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{13}{30} a^{4} + \frac{7}{30} a^{3} - \frac{7}{15} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{24630} a^{18} - \frac{193}{24630} a^{17} + \frac{42}{4105} a^{16} + \frac{601}{12315} a^{15} + \frac{283}{24630} a^{14} - \frac{296}{12315} a^{13} - \frac{337}{12315} a^{12} - \frac{377}{8210} a^{11} + \frac{553}{12315} a^{10} - \frac{638}{12315} a^{9} - \frac{311}{4926} a^{8} - \frac{2639}{12315} a^{7} - \frac{4378}{12315} a^{6} - \frac{1551}{8210} a^{5} + \frac{1904}{12315} a^{4} + \frac{11911}{24630} a^{3} - \frac{365}{2463} a^{2} - \frac{1120}{2463} a - \frac{898}{2463}$, $\frac{1}{872369970} a^{19} + \frac{1493}{290789990} a^{18} + \frac{4445722}{436184985} a^{17} + \frac{10215293}{872369970} a^{16} + \frac{11735503}{290789990} a^{15} + \frac{2135893}{290789990} a^{14} + \frac{1380739}{290789990} a^{13} - \frac{40966961}{872369970} a^{12} + \frac{37482143}{872369970} a^{11} - \frac{55598717}{872369970} a^{10} + \frac{318335257}{872369970} a^{9} - \frac{243523661}{872369970} a^{8} + \frac{19500183}{290789990} a^{7} - \frac{56383447}{174473994} a^{6} - \frac{426359111}{872369970} a^{5} + \frac{73849522}{436184985} a^{4} - \frac{93051917}{436184985} a^{3} - \frac{8050017}{290789990} a^{2} - \frac{38038235}{87236997} a - \frac{27212191}{87236997}$

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:  \( -\frac{8920766}{436184985} a^{19} + \frac{9795325}{87236997} a^{18} - \frac{126177334}{436184985} a^{17} + \frac{226700252}{436184985} a^{16} - \frac{111202304}{145394995} a^{15} + \frac{143515338}{145394995} a^{14} - \frac{453004468}{436184985} a^{13} + \frac{377269298}{436184985} a^{12} - \frac{78661150}{87236997} a^{11} + \frac{151215460}{87236997} a^{10} - \frac{1345027934}{436184985} a^{9} + \frac{343015658}{87236997} a^{8} - \frac{2459019056}{436184985} a^{7} + \frac{1245134046}{145394995} a^{6} - \frac{3894800978}{436184985} a^{5} + \frac{714388962}{145394995} a^{4} - \frac{151277444}{145394995} a^{3} + \frac{532664932}{436184985} a^{2} - \frac{104478484}{87236997} a + \frac{31399886}{87236997} \) (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:  \( 9032.09324455 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.171125.1, 10.2.146418828125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$