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Properties

Label 20.12.3009103398...3776.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 11^{18}\cdot 727^{2}$
Root discriminant $23.65$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -20, 55, 84, -311, -217, 1089, -61, -1704, 692, 1047, -442, -308, -51, 168, -3, -11, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 - 11*x^17 - 3*x^16 + 168*x^15 - 51*x^14 - 308*x^13 - 442*x^12 + 1047*x^11 + 692*x^10 - 1704*x^9 - 61*x^8 + 1089*x^7 - 217*x^6 - 311*x^5 + 84*x^4 + 55*x^3 - 20*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 5*x^18 - 11*x^17 - 3*x^16 + 168*x^15 - 51*x^14 - 308*x^13 - 442*x^12 + 1047*x^11 + 692*x^10 - 1704*x^9 - 61*x^8 + 1089*x^7 - 217*x^6 - 311*x^5 + 84*x^4 + 55*x^3 - 20*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 5 x^{18} - 11 x^{17} - 3 x^{16} + 168 x^{15} - 51 x^{14} - 308 x^{13} - 442 x^{12} + 1047 x^{11} + 692 x^{10} - 1704 x^{9} - 61 x^{8} + 1089 x^{7} - 217 x^{6} - 311 x^{5} + 84 x^{4} + 55 x^{3} - 20 x^{2} - 2 x + 1 \)

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:  \(3009103398689521267119563776=2^{10}\cdot 11^{18}\cdot 727^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.65$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3818340508460873} a^{19} - \frac{1073459284459088}{3818340508460873} a^{18} + \frac{66348323716796}{3818340508460873} a^{17} - \frac{316170435095717}{3818340508460873} a^{16} + \frac{1839774131603404}{3818340508460873} a^{15} + \frac{1604346135717051}{3818340508460873} a^{14} - \frac{374530062272077}{3818340508460873} a^{13} - \frac{124528292343965}{3818340508460873} a^{12} + \frac{850389371016462}{3818340508460873} a^{11} + \frac{1825975119380480}{3818340508460873} a^{10} - \frac{1028927647993839}{3818340508460873} a^{9} - \frac{1065221542474185}{3818340508460873} a^{8} + \frac{182404773713738}{3818340508460873} a^{7} + \frac{60917097893618}{3818340508460873} a^{6} - \frac{1426946080685589}{3818340508460873} a^{5} - \frac{1868094901015580}{3818340508460873} a^{4} + \frac{1288949011451275}{3818340508460873} a^{3} - \frac{655430613397278}{3818340508460873} a^{2} + \frac{1881375418643129}{3818340508460873} a - \frac{1697311228692833}{3818340508460873}$

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:  \( 2785954.7019 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
727Data not computed