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Properties

Label 20.20.1935503049...6289.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 11^{16}\cdot 61^{10}$
Root discriminant $92.12$
Ramified primes $3, 11, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times A_4$ (as 20T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2707, 6887, -25936, -62457, 94551, 198770, -172211, -285884, 174189, 213724, -101908, -88642, 35038, 20694, -7020, -2640, 788, 167, -45, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 45*x^18 + 167*x^17 + 788*x^16 - 2640*x^15 - 7020*x^14 + 20694*x^13 + 35038*x^12 - 88642*x^11 - 101908*x^10 + 213724*x^9 + 174189*x^8 - 285884*x^7 - 172211*x^6 + 198770*x^5 + 94551*x^4 - 62457*x^3 - 25936*x^2 + 6887*x + 2707)
 
gp: K = bnfinit(x^20 - 4*x^19 - 45*x^18 + 167*x^17 + 788*x^16 - 2640*x^15 - 7020*x^14 + 20694*x^13 + 35038*x^12 - 88642*x^11 - 101908*x^10 + 213724*x^9 + 174189*x^8 - 285884*x^7 - 172211*x^6 + 198770*x^5 + 94551*x^4 - 62457*x^3 - 25936*x^2 + 6887*x + 2707, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 45 x^{18} + 167 x^{17} + 788 x^{16} - 2640 x^{15} - 7020 x^{14} + 20694 x^{13} + 35038 x^{12} - 88642 x^{11} - 101908 x^{10} + 213724 x^{9} + 174189 x^{8} - 285884 x^{7} - 172211 x^{6} + 198770 x^{5} + 94551 x^{4} - 62457 x^{3} - 25936 x^{2} + 6887 x + 2707 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1935503049159664173634294979108883566289=3^{10}\cdot 11^{16}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.12$
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:  $3, 11, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{19476694796975112694222635810175576681} a^{19} + \frac{442292413833000358244173986113355104}{19476694796975112694222635810175576681} a^{18} + \frac{488907766068440401437457261992336427}{19476694796975112694222635810175576681} a^{17} - \frac{1030400764838669245124910730142989226}{19476694796975112694222635810175576681} a^{16} - \frac{8947452671455968802051975419373343625}{19476694796975112694222635810175576681} a^{15} + \frac{5999179404544121769197415048387993383}{19476694796975112694222635810175576681} a^{14} - \frac{1154519781062948455629553878947764879}{19476694796975112694222635810175576681} a^{13} - \frac{9552164113296614767921262488560261568}{19476694796975112694222635810175576681} a^{12} - \frac{186340094308178601129168433609315265}{19476694796975112694222635810175576681} a^{11} + \frac{1343466617587488238221950785759659954}{19476694796975112694222635810175576681} a^{10} - \frac{2368602152739203355585705876027602535}{19476694796975112694222635810175576681} a^{9} - \frac{4513641199466935118203089488219735505}{19476694796975112694222635810175576681} a^{8} + \frac{6981588297644171575027754357670047269}{19476694796975112694222635810175576681} a^{7} + \frac{93871054131695355972636989002681166}{19476694796975112694222635810175576681} a^{6} - \frac{5040176764190145696134558278530325199}{19476694796975112694222635810175576681} a^{5} + \frac{8633333430286364929148708035555624032}{19476694796975112694222635810175576681} a^{4} + \frac{198299857595211655351351567884576509}{19476694796975112694222635810175576681} a^{3} + \frac{6477980682208320213327191144227461585}{19476694796975112694222635810175576681} a^{2} - \frac{2342121471056691230809701688883728361}{19476694796975112694222635810175576681} a + \frac{7630101764573699630191663972765426336}{19476694796975112694222635810175576681}$

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:  $19$
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:  \( 13755912027100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times A_4$ (as 20T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 60
The 20 conjugacy class representatives for $C_5\times A_4$
Character table for $C_5\times A_4$

Intermediate fields

4.4.33489.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 30 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
3Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
61Data not computed