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Properties

Label 20.8.30091033986...3776.1
Degree $20$
Signature $[8, 6]$
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![-43, 17, 304, -275, -742, 1081, 603, -1727, 135, 1646, -892, -1062, 1632, -693, -215, 360, -141, 0, 19, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 19*x^18 - 141*x^16 + 360*x^15 - 215*x^14 - 693*x^13 + 1632*x^12 - 1062*x^11 - 892*x^10 + 1646*x^9 + 135*x^8 - 1727*x^7 + 603*x^6 + 1081*x^5 - 742*x^4 - 275*x^3 + 304*x^2 + 17*x - 43)
 
gp: K = bnfinit(x^20 - 7*x^19 + 19*x^18 - 141*x^16 + 360*x^15 - 215*x^14 - 693*x^13 + 1632*x^12 - 1062*x^11 - 892*x^10 + 1646*x^9 + 135*x^8 - 1727*x^7 + 603*x^6 + 1081*x^5 - 742*x^4 - 275*x^3 + 304*x^2 + 17*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 19 x^{18} - 141 x^{16} + 360 x^{15} - 215 x^{14} - 693 x^{13} + 1632 x^{12} - 1062 x^{11} - 892 x^{10} + 1646 x^{9} + 135 x^{8} - 1727 x^{7} + 603 x^{6} + 1081 x^{5} - 742 x^{4} - 275 x^{3} + 304 x^{2} + 17 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{109771732008204089561} a^{19} - \frac{50398482994754451181}{109771732008204089561} a^{18} + \frac{20360234214684381280}{109771732008204089561} a^{17} + \frac{30475314237365252295}{109771732008204089561} a^{16} + \frac{18208755162652978302}{109771732008204089561} a^{15} + \frac{36924999183270332121}{109771732008204089561} a^{14} - \frac{50755244511630038456}{109771732008204089561} a^{13} + \frac{45023320833424291870}{109771732008204089561} a^{12} + \frac{41678838661740177585}{109771732008204089561} a^{11} + \frac{44098143374171935529}{109771732008204089561} a^{10} - \frac{37940973191059204919}{109771732008204089561} a^{9} - \frac{21716828034673278536}{109771732008204089561} a^{8} - \frac{20470621887627233046}{109771732008204089561} a^{7} + \frac{25635563101066957818}{109771732008204089561} a^{6} + \frac{4113581381517008511}{109771732008204089561} a^{5} + \frac{12152297536063754035}{109771732008204089561} a^{4} - \frac{1206790947702963195}{109771732008204089561} a^{3} - \frac{2157119282087975128}{109771732008204089561} a^{2} + \frac{6116710951604959495}{109771732008204089561} a - \frac{1168595974469461111}{2552830976934978827}$

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:  $13$
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:  \( 1231106.23342 \) (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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$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