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Properties

Label 20.14.5140615083...6064.2
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{46}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $48.48$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -128, -1032, 160, 14584, 648, -49614, 1444, 60204, -6536, -29132, 8088, 6790, -2868, -1179, 302, 213, -4, -19, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 19*x^18 - 4*x^17 + 213*x^16 + 302*x^15 - 1179*x^14 - 2868*x^13 + 6790*x^12 + 8088*x^11 - 29132*x^10 - 6536*x^9 + 60204*x^8 + 1444*x^7 - 49614*x^6 + 648*x^5 + 14584*x^4 + 160*x^3 - 1032*x^2 - 128*x - 4)
 
gp: K = bnfinit(x^20 - 2*x^19 - 19*x^18 - 4*x^17 + 213*x^16 + 302*x^15 - 1179*x^14 - 2868*x^13 + 6790*x^12 + 8088*x^11 - 29132*x^10 - 6536*x^9 + 60204*x^8 + 1444*x^7 - 49614*x^6 + 648*x^5 + 14584*x^4 + 160*x^3 - 1032*x^2 - 128*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 19 x^{18} - 4 x^{17} + 213 x^{16} + 302 x^{15} - 1179 x^{14} - 2868 x^{13} + 6790 x^{12} + 8088 x^{11} - 29132 x^{10} - 6536 x^{9} + 60204 x^{8} + 1444 x^{7} - 49614 x^{6} + 648 x^{5} + 14584 x^{4} + 160 x^{3} - 1032 x^{2} - 128 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5140615083075201996699936766296064=-\,2^{46}\cdot 31^{7}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.48$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6}$, $\frac{1}{469896622688846169507989686777026243604} a^{19} + \frac{22089168717562769379095745646668859779}{234948311344423084753994843388513121802} a^{18} - \frac{8844589865951117443341588923859142273}{234948311344423084753994843388513121802} a^{17} + \frac{21893649140231891520240998683290677527}{234948311344423084753994843388513121802} a^{16} - \frac{109550193582495958423960263730923868103}{469896622688846169507989686777026243604} a^{15} - \frac{7659284312984858135334570362591533834}{117474155672211542376997421694256560901} a^{14} + \frac{10471508712068355698018598615891572251}{234948311344423084753994843388513121802} a^{13} + \frac{22579649424823586143303707208973722743}{234948311344423084753994843388513121802} a^{12} + \frac{8687064319347600538284422255801714826}{117474155672211542376997421694256560901} a^{11} - \frac{97355161216150841968253437246851408015}{234948311344423084753994843388513121802} a^{10} - \frac{102002196134993153226148145148636961197}{234948311344423084753994843388513121802} a^{9} + \frac{10153636923482152959339594218492351439}{234948311344423084753994843388513121802} a^{8} + \frac{31543029753106141725778260096813242225}{234948311344423084753994843388513121802} a^{7} - \frac{6802036864874009553417624131428946431}{117474155672211542376997421694256560901} a^{6} - \frac{25830074676669562039458463376427892474}{117474155672211542376997421694256560901} a^{5} + \frac{49440691205933973543385423123165877579}{117474155672211542376997421694256560901} a^{4} - \frac{25555401823145914175937210386093983278}{117474155672211542376997421694256560901} a^{3} - \frac{14865416150777966873532587282694606653}{117474155672211542376997421694256560901} a^{2} + \frac{41200483409595525017456854816522944951}{117474155672211542376997421694256560901} a - \frac{38190319053635922657856854489282748398}{117474155672211542376997421694256560901}$

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:  $16$
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:  \( 51834053863.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.8.24.7$x^{8} + 8 x^{7} + 12 x^{6} + 10 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
31Data not computed
227Data not computed