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Properties

Label 20.10.3611383131...7259.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{10}\cdot 11^{19}$
Root discriminant $16.90$
Ramified primes $3, 11$
Class number $1$
Class group Trivial
Galois group $C_5\times D_4$ (as 20T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 5, -46, -145, -71, 329, 490, -208, -796, -33, 613, 37, -191, 17, -27, 4, 25, -7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 7*x^18 + 25*x^17 + 4*x^16 - 27*x^15 + 17*x^14 - 191*x^13 + 37*x^12 + 613*x^11 - 33*x^10 - 796*x^9 - 208*x^8 + 490*x^7 + 329*x^6 - 71*x^5 - 145*x^4 - 46*x^3 + 5*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 7*x^18 + 25*x^17 + 4*x^16 - 27*x^15 + 17*x^14 - 191*x^13 + 37*x^12 + 613*x^11 - 33*x^10 - 796*x^9 - 208*x^8 + 490*x^7 + 329*x^6 - 71*x^5 - 145*x^4 - 46*x^3 + 5*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 7 x^{18} + 25 x^{17} + 4 x^{16} - 27 x^{15} + 17 x^{14} - 191 x^{13} + 37 x^{12} + 613 x^{11} - 33 x^{10} - 796 x^{9} - 208 x^{8} + 490 x^{7} + 329 x^{6} - 71 x^{5} - 145 x^{4} - 46 x^{3} + 5 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3611383131888430543937259=-\,3^{10}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.90$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{666666622348781} a^{19} - \frac{122509765371255}{666666622348781} a^{18} - \frac{56231766196917}{666666622348781} a^{17} - \frac{320116522375897}{666666622348781} a^{16} + \frac{166129933929497}{666666622348781} a^{15} + \frac{42733504934954}{666666622348781} a^{14} - \frac{66965778082646}{666666622348781} a^{13} - \frac{313668546203612}{666666622348781} a^{12} + \frac{61297539816898}{666666622348781} a^{11} - \frac{67157009711282}{666666622348781} a^{10} + \frac{180507555031729}{666666622348781} a^{9} - \frac{54036129502594}{666666622348781} a^{8} + \frac{258185973769170}{666666622348781} a^{7} - \frac{279020065656553}{666666622348781} a^{6} + \frac{127881037895927}{666666622348781} a^{5} + \frac{131627135947625}{666666622348781} a^{4} - \frac{213071899651706}{666666622348781} a^{3} + \frac{44319020621294}{666666622348781} a^{2} + \frac{109932448092360}{666666622348781} a - \frac{281968307593006}{666666622348781}$

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:  $14$
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:  \( 54412.0450771 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times D_4$ (as 20T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 25 conjugacy class representatives for $C_5\times D_4$
Character table for $C_5\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), 4.2.11979.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
11Data not computed