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Properties

Label 20.20.1817615687...2833.1
Degree $20$
Signature $[20, 0]$
Discriminant $61^{9}\cdot 397^{7}$
Root discriminant $51.64$
Ramified primes $61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_5$ (as 20T65)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, -3763, -14297, 16362, 66241, -33440, -126766, 41523, 130714, -32462, -79106, 15285, 28833, -3987, -6277, 505, 770, -22, -46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 46*x^18 - 22*x^17 + 770*x^16 + 505*x^15 - 6277*x^14 - 3987*x^13 + 28833*x^12 + 15285*x^11 - 79106*x^10 - 32462*x^9 + 130714*x^8 + 41523*x^7 - 126766*x^6 - 33440*x^5 + 66241*x^4 + 16362*x^3 - 14297*x^2 - 3763*x - 67)
 
gp: K = bnfinit(x^20 - 46*x^18 - 22*x^17 + 770*x^16 + 505*x^15 - 6277*x^14 - 3987*x^13 + 28833*x^12 + 15285*x^11 - 79106*x^10 - 32462*x^9 + 130714*x^8 + 41523*x^7 - 126766*x^6 - 33440*x^5 + 66241*x^4 + 16362*x^3 - 14297*x^2 - 3763*x - 67, 1)
 

Normalized defining polynomial

\( x^{20} - 46 x^{18} - 22 x^{17} + 770 x^{16} + 505 x^{15} - 6277 x^{14} - 3987 x^{13} + 28833 x^{12} + 15285 x^{11} - 79106 x^{10} - 32462 x^{9} + 130714 x^{8} + 41523 x^{7} - 126766 x^{6} - 33440 x^{5} + 66241 x^{4} + 16362 x^{3} - 14297 x^{2} - 3763 x - 67 \)

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:  \(18176156878239859979587199616662833=61^{9}\cdot 397^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.64$
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:  $61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{397} a^{18} + \frac{36}{397} a^{17} + \frac{132}{397} a^{16} - \frac{185}{397} a^{15} + \frac{173}{397} a^{14} - \frac{23}{397} a^{13} - \frac{34}{397} a^{12} - \frac{141}{397} a^{11} - \frac{163}{397} a^{10} - \frac{82}{397} a^{9} + \frac{132}{397} a^{8} + \frac{49}{397} a^{7} - \frac{12}{397} a^{6} - \frac{193}{397} a^{5} - \frac{7}{397} a^{4} - \frac{141}{397} a^{3} - \frac{87}{397} a^{2} + \frac{158}{397} a + \frac{126}{397}$, $\frac{1}{38708404961621482738817} a^{19} - \frac{42200915080361033814}{38708404961621482738817} a^{18} - \frac{10733015825945963943640}{38708404961621482738817} a^{17} + \frac{8173103930132821209971}{38708404961621482738817} a^{16} + \frac{9171096908472860931182}{38708404961621482738817} a^{15} - \frac{8613982696331597548804}{38708404961621482738817} a^{14} + \frac{18714671754948244212884}{38708404961621482738817} a^{13} + \frac{11832251337795164103929}{38708404961621482738817} a^{12} + \frac{2353269386364047847477}{38708404961621482738817} a^{11} - \frac{18437918832158552285126}{38708404961621482738817} a^{10} + \frac{7888700736176874493395}{38708404961621482738817} a^{9} + \frac{3627805651096575418994}{38708404961621482738817} a^{8} - \frac{16193878710823178367146}{38708404961621482738817} a^{7} - \frac{4855567087528718609259}{38708404961621482738817} a^{6} + \frac{7093744119782911077570}{38708404961621482738817} a^{5} - \frac{7476828809755423671154}{38708404961621482738817} a^{4} + \frac{14803517630333137404649}{38708404961621482738817} a^{3} - \frac{7107250655029391104493}{38708404961621482738817} a^{2} - \frac{18685670695320543514555}{38708404961621482738817} a - \frac{1629314822031957402652}{38708404961621482738817}$

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:  \( 47567559110.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T65):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

10.10.14202376626313.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
61Data not computed
397Data not computed