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Properties

Label 20.6.83510262074...0187.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 347^{7}$
Root discriminant $27.93$
Ramified primes $13, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-61, 1, 2869, -12047, 18605, -10785, 876, 4040, -7493, 5936, -1727, -1364, 1656, -663, -36, 158, -83, 13, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 6*x^18 + 13*x^17 - 83*x^16 + 158*x^15 - 36*x^14 - 663*x^13 + 1656*x^12 - 1364*x^11 - 1727*x^10 + 5936*x^9 - 7493*x^8 + 4040*x^7 + 876*x^6 - 10785*x^5 + 18605*x^4 - 12047*x^3 + 2869*x^2 + x - 61)
 
gp: K = bnfinit(x^20 - 3*x^19 + 6*x^18 + 13*x^17 - 83*x^16 + 158*x^15 - 36*x^14 - 663*x^13 + 1656*x^12 - 1364*x^11 - 1727*x^10 + 5936*x^9 - 7493*x^8 + 4040*x^7 + 876*x^6 - 10785*x^5 + 18605*x^4 - 12047*x^3 + 2869*x^2 + x - 61, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 6 x^{18} + 13 x^{17} - 83 x^{16} + 158 x^{15} - 36 x^{14} - 663 x^{13} + 1656 x^{12} - 1364 x^{11} - 1727 x^{10} + 5936 x^{9} - 7493 x^{8} + 4040 x^{7} + 876 x^{6} - 10785 x^{5} + 18605 x^{4} - 12047 x^{3} + 2869 x^{2} + x - 61 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-83510262074750967216189850187=-\,13^{10}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.93$
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:  $13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2761} a^{18} - \frac{181}{2761} a^{17} + \frac{908}{2761} a^{16} + \frac{1151}{2761} a^{15} - \frac{36}{2761} a^{14} + \frac{1183}{2761} a^{13} + \frac{114}{2761} a^{12} - \frac{1358}{2761} a^{11} + \frac{361}{2761} a^{10} + \frac{87}{2761} a^{9} + \frac{52}{251} a^{8} + \frac{1370}{2761} a^{7} + \frac{514}{2761} a^{6} + \frac{1159}{2761} a^{5} - \frac{906}{2761} a^{4} - \frac{510}{2761} a^{3} - \frac{795}{2761} a^{2} + \frac{112}{251} a - \frac{786}{2761}$, $\frac{1}{1312515329409926775002047772907203} a^{19} + \frac{12958278314428562516494380502}{1312515329409926775002047772907203} a^{18} - \frac{33635903951172930529252468199141}{1312515329409926775002047772907203} a^{17} - \frac{116107437184265659908283186196489}{1312515329409926775002047772907203} a^{16} - \frac{423515973614275608483398770494412}{1312515329409926775002047772907203} a^{15} - \frac{109143660713743251778928792420335}{1312515329409926775002047772907203} a^{14} + \frac{494827400450823621995010102596365}{1312515329409926775002047772907203} a^{13} - \frac{616808059850247578946989523373222}{1312515329409926775002047772907203} a^{12} - \frac{467087466797397806000576254469163}{1312515329409926775002047772907203} a^{11} + \frac{526353661480069433927452257687162}{1312515329409926775002047772907203} a^{10} - \frac{30531086249140301414718667812361}{119319575400902434091095252082473} a^{9} - \frac{412930612781340789798709946862000}{1312515329409926775002047772907203} a^{8} - \frac{379038055505480603061362453763274}{1312515329409926775002047772907203} a^{7} + \frac{157497646929469260620378893339722}{1312515329409926775002047772907203} a^{6} + \frac{557036163287883189907635676229364}{1312515329409926775002047772907203} a^{5} - \frac{437903841937447524188411848247287}{1312515329409926775002047772907203} a^{4} + \frac{632427414544629309227624904306325}{1312515329409926775002047772907203} a^{3} - \frac{26705897238121712423339646774739}{119319575400902434091095252082473} a^{2} - \frac{235203362718297068111915568921630}{1312515329409926775002047772907203} a - \frac{3017036864410177874869989274440}{119319575400902434091095252082473}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $12$
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:  \( 4572338.49147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.2.58643.1, 5.3.4511.1, 10.6.44707018837.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 siblings: data not computed
Degree 40 siblings: 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed