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Properties

Label 20.0.59699157574...1344.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}$
Root discriminant $24.48$
Ramified primes $2, 11$
Class number $2$
Class group $[2]$
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -18, 132, -438, 591, 286, -1580, 522, 2255, -1752, -1288, 1888, 44, -948, 322, 198, -147, 6, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 22 x^{18} + 6 x^{17} - 147 x^{16} + 198 x^{15} + 322 x^{14} - 948 x^{13} + 44 x^{12} + 1888 x^{11} - 1288 x^{10} - 1752 x^{9} + 2255 x^{8} + 522 x^{7} - 1580 x^{6} + 286 x^{5} + 591 x^{4} - 438 x^{3} + 132 x^{2} - 18 x + 1 \)

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:  \(5969915757478328440239161344=2^{30}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.48$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13427801759600274892759} a^{19} + \frac{2807273846780086075493}{13427801759600274892759} a^{18} + \frac{6556186455706833329319}{13427801759600274892759} a^{17} - \frac{3437764703720670071482}{13427801759600274892759} a^{16} - \frac{38813976626290479847}{13427801759600274892759} a^{15} - \frac{1545668896118073996327}{13427801759600274892759} a^{14} + \frac{2584438624244546471218}{13427801759600274892759} a^{13} - \frac{1960808599917551822194}{13427801759600274892759} a^{12} - \frac{1182663511097494624786}{13427801759600274892759} a^{11} + \frac{2289370007905317897449}{13427801759600274892759} a^{10} + \frac{5965873348852191194379}{13427801759600274892759} a^{9} - \frac{3746375885629977389445}{13427801759600274892759} a^{8} + \frac{5133315959021649426850}{13427801759600274892759} a^{7} + \frac{519817447431889616049}{13427801759600274892759} a^{6} + \frac{176240201811364473633}{583817467808707604033} a^{5} + \frac{5695813971918982193084}{13427801759600274892759} a^{4} - \frac{1500836676219838062879}{13427801759600274892759} a^{3} + \frac{4465967603096377461673}{13427801759600274892759} a^{2} + \frac{4282834874868009869364}{13427801759600274892759} a + \frac{174925273020673232102}{13427801759600274892759}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 159636.716542 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.2414538435584.1, 10.2.2414538435584.1, 10.4.219503494144.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Arithmetically equvalently 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
11Data not computed