↑

Properties

Label 20.12.7811260074...0625.2
Degree $20$
Signature $[12, 4]$
Discriminant $5^{14}\cdot 61^{6}\cdot 397^{4}$
Root discriminant $35.05$
Ramified primes $5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![499, -1238, -10588, 20878, 11265, -33067, 14589, 3706, -18233, 13464, -2169, -2543, 3791, -1836, 494, 5, -103, 32, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 7*x^18 + 32*x^17 - 103*x^16 + 5*x^15 + 494*x^14 - 1836*x^13 + 3791*x^12 - 2543*x^11 - 2169*x^10 + 13464*x^9 - 18233*x^8 + 3706*x^7 + 14589*x^6 - 33067*x^5 + 11265*x^4 + 20878*x^3 - 10588*x^2 - 1238*x + 499)
 
gp: K = bnfinit(x^20 - x^19 - 7*x^18 + 32*x^17 - 103*x^16 + 5*x^15 + 494*x^14 - 1836*x^13 + 3791*x^12 - 2543*x^11 - 2169*x^10 + 13464*x^9 - 18233*x^8 + 3706*x^7 + 14589*x^6 - 33067*x^5 + 11265*x^4 + 20878*x^3 - 10588*x^2 - 1238*x + 499, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 7 x^{18} + 32 x^{17} - 103 x^{16} + 5 x^{15} + 494 x^{14} - 1836 x^{13} + 3791 x^{12} - 2543 x^{11} - 2169 x^{10} + 13464 x^{9} - 18233 x^{8} + 3706 x^{7} + 14589 x^{6} - 33067 x^{5} + 11265 x^{4} + 20878 x^{3} - 10588 x^{2} - 1238 x + 499 \)

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:  \(7811260074827935595953369140625=5^{14}\cdot 61^{6}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.05$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{10239318760283560220856977119148278520297} a^{19} - \frac{740271877121213574636896979559587934863}{10239318760283560220856977119148278520297} a^{18} + \frac{392008644573245495934509183971178835547}{10239318760283560220856977119148278520297} a^{17} - \frac{3377430486034732716416350721618100270032}{10239318760283560220856977119148278520297} a^{16} - \frac{4954945807805175720436563887386108993427}{10239318760283560220856977119148278520297} a^{15} - \frac{523128308380232072911414023465556040545}{10239318760283560220856977119148278520297} a^{14} - \frac{1820635690343884894832118128923266564397}{10239318760283560220856977119148278520297} a^{13} + \frac{795456859048722471362633165576407339072}{10239318760283560220856977119148278520297} a^{12} - \frac{3155687060046939725989850192937162623944}{10239318760283560220856977119148278520297} a^{11} + \frac{2324014466011091132704177684045138411459}{10239318760283560220856977119148278520297} a^{10} + \frac{4022271936245307590406271548615941446461}{10239318760283560220856977119148278520297} a^{9} - \frac{328333306644127931288981540900241807231}{10239318760283560220856977119148278520297} a^{8} - \frac{305741466352815744087318919325643125436}{10239318760283560220856977119148278520297} a^{7} - \frac{1422134058713724482217482082010461048166}{10239318760283560220856977119148278520297} a^{6} - \frac{2731738494206908128731142205350528926200}{10239318760283560220856977119148278520297} a^{5} + \frac{1481690513411723530609487506551684256628}{10239318760283560220856977119148278520297} a^{4} + \frac{4239616580817952786835475002131126711866}{10239318760283560220856977119148278520297} a^{3} + \frac{1054935741164669496179634722489761611346}{10239318760283560220856977119148278520297} a^{2} + \frac{10624685006565547406735611048809711571}{10239318760283560220856977119148278520297} a + \frac{2948744138897025131692779480023602953154}{10239318760283560220856977119148278520297}$

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:  \( 141876204.164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
5Data not computed
61Data not computed
397Data not computed