↑

Properties

Label 20.2.79411198040...9936.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{40}\cdot 13^{12}\cdot 31$
Root discriminant $22.13$
Ramified primes $2, 13, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 12, -34, 48, -12, -104, 284, -544, 844, -1012, 1004, -872, 554, -164, -2, -36, 48, -8, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 - 8*x^17 + 48*x^16 - 36*x^15 - 2*x^14 - 164*x^13 + 554*x^12 - 872*x^11 + 1004*x^10 - 1012*x^9 + 844*x^8 - 544*x^7 + 284*x^6 - 104*x^5 - 12*x^4 + 48*x^3 - 34*x^2 + 12*x - 2)
 
gp: K = bnfinit(x^20 - 6*x^18 - 8*x^17 + 48*x^16 - 36*x^15 - 2*x^14 - 164*x^13 + 554*x^12 - 872*x^11 + 1004*x^10 - 1012*x^9 + 844*x^8 - 544*x^7 + 284*x^6 - 104*x^5 - 12*x^4 + 48*x^3 - 34*x^2 + 12*x - 2, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} - 8 x^{17} + 48 x^{16} - 36 x^{15} - 2 x^{14} - 164 x^{13} + 554 x^{12} - 872 x^{11} + 1004 x^{10} - 1012 x^{9} + 844 x^{8} - 544 x^{7} + 284 x^{6} - 104 x^{5} - 12 x^{4} + 48 x^{3} - 34 x^{2} + 12 x - 2 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-794111980409569671890599936=-\,2^{40}\cdot 13^{12}\cdot 31\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.13$
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:  $2, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4408679991153049} a^{19} + \frac{1619926101317389}{4408679991153049} a^{18} + \frac{360429318782755}{4408679991153049} a^{17} - \frac{706220112383938}{4408679991153049} a^{16} + \frac{998409021103636}{4408679991153049} a^{15} - \frac{1986394181458615}{4408679991153049} a^{14} + \frac{610702053863073}{4408679991153049} a^{13} - \frac{978504657232603}{4408679991153049} a^{12} - \frac{467395788418056}{4408679991153049} a^{11} - \frac{166764988413113}{4408679991153049} a^{10} - \frac{702282777288613}{4408679991153049} a^{9} + \frac{1440739133323094}{4408679991153049} a^{8} - \frac{1831364004084930}{4408679991153049} a^{7} + \frac{598939679974429}{4408679991153049} a^{6} - \frac{1219048404829199}{4408679991153049} a^{5} - \frac{1240921892144950}{4408679991153049} a^{4} - \frac{1576305304755981}{4408679991153049} a^{3} + \frac{1870939078933931}{4408679991153049} a^{2} - \frac{1705138697104542}{4408679991153049} a - \frac{2075260925149780}{4408679991153049}$

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:  $10$
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:  \( 519411.221896 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.1.35152.1, 10.2.2530638036992.1

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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
13Data not computed
31Data not computed