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Properties

Label 20.12.4933814675...1664.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{30}\cdot 11^{16}$
Root discriminant $19.26$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 22, -158, 542, -1042, 564, 1784, -3458, 2214, -866, 906, -362, -577, 658, -296, 76, 14, -40, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 26*x^18 - 40*x^17 + 14*x^16 + 76*x^15 - 296*x^14 + 658*x^13 - 577*x^12 - 362*x^11 + 906*x^10 - 866*x^9 + 2214*x^8 - 3458*x^7 + 1784*x^6 + 564*x^5 - 1042*x^4 + 542*x^3 - 158*x^2 + 22*x - 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 26*x^18 - 40*x^17 + 14*x^16 + 76*x^15 - 296*x^14 + 658*x^13 - 577*x^12 - 362*x^11 + 906*x^10 - 866*x^9 + 2214*x^8 - 3458*x^7 + 1784*x^6 + 564*x^5 - 1042*x^4 + 542*x^3 - 158*x^2 + 22*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 26 x^{18} - 40 x^{17} + 14 x^{16} + 76 x^{15} - 296 x^{14} + 658 x^{13} - 577 x^{12} - 362 x^{11} + 906 x^{10} - 866 x^{9} + 2214 x^{8} - 3458 x^{7} + 1784 x^{6} + 564 x^{5} - 1042 x^{4} + 542 x^{3} - 158 x^{2} + 22 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(49338146756019243307761664=2^{30}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.26$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{281159511642280259} a^{19} - \frac{790872066861048}{281159511642280259} a^{18} + \frac{75349649291984510}{281159511642280259} a^{17} - \frac{132263796683276324}{281159511642280259} a^{16} - \frac{94228999132986126}{281159511642280259} a^{15} - \frac{124662821699652005}{281159511642280259} a^{14} + \frac{66794819680697446}{281159511642280259} a^{13} - \frac{8687836607352692}{281159511642280259} a^{12} - \frac{788109571468196}{281159511642280259} a^{11} + \frac{13346611640413440}{281159511642280259} a^{10} - \frac{1680799411687558}{281159511642280259} a^{9} + \frac{131608348234449829}{281159511642280259} a^{8} - \frac{26666918778302790}{281159511642280259} a^{7} - \frac{117904264968911291}{281159511642280259} a^{6} - \frac{125040291466360622}{281159511642280259} a^{5} - \frac{22505162673965665}{281159511642280259} a^{4} - \frac{33487978768389524}{281159511642280259} a^{3} + \frac{110366717884731243}{281159511642280259} a^{2} + \frac{126918426019515797}{281159511642280259} a + \frac{19814846242433604}{281159511642280259}$

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:  $15$
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:  \( 311138.716179 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1 x2, 10.6.219503494144.2

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 20 siblings: data not computed
Degree 32 sibling: 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 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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$