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Properties

Label 20.12.1606056182...5712.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{34}\cdot 2657^{7}$
Root discriminant $51.32$
Ramified primes $2, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7081, -72472, 234632, -209264, -164472, 197948, 315634, -443464, -6853, 271060, -126598, -40956, 45225, -2892, -5592, 500, 522, -12, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 36*x^18 - 12*x^17 + 522*x^16 + 500*x^15 - 5592*x^14 - 2892*x^13 + 45225*x^12 - 40956*x^11 - 126598*x^10 + 271060*x^9 - 6853*x^8 - 443464*x^7 + 315634*x^6 + 197948*x^5 - 164472*x^4 - 209264*x^3 + 234632*x^2 - 72472*x + 7081)
 
gp: K = bnfinit(x^20 - 36*x^18 - 12*x^17 + 522*x^16 + 500*x^15 - 5592*x^14 - 2892*x^13 + 45225*x^12 - 40956*x^11 - 126598*x^10 + 271060*x^9 - 6853*x^8 - 443464*x^7 + 315634*x^6 + 197948*x^5 - 164472*x^4 - 209264*x^3 + 234632*x^2 - 72472*x + 7081, 1)
 

Normalized defining polynomial

\( x^{20} - 36 x^{18} - 12 x^{17} + 522 x^{16} + 500 x^{15} - 5592 x^{14} - 2892 x^{13} + 45225 x^{12} - 40956 x^{11} - 126598 x^{10} + 271060 x^{9} - 6853 x^{8} - 443464 x^{7} + 315634 x^{6} + 197948 x^{5} - 164472 x^{4} - 209264 x^{3} + 234632 x^{2} - 72472 x + 7081 \)

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:  \(16060561821304241463001418628595712=2^{34}\cdot 2657^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.32$
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:  $2, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1067883344250410135420675984725271941866808158308} a^{19} - \frac{24420779121591154446528084147429528783610979367}{533941672125205067710337992362635970933404079154} a^{18} + \frac{46437900947786806900269335372327300871251986027}{1067883344250410135420675984725271941866808158308} a^{17} - \frac{8773074308412952143753638827362131130088729913}{266970836062602533855168996181317985466702039577} a^{16} + \frac{82442390571874700316344301374256222506911796263}{1067883344250410135420675984725271941866808158308} a^{15} + \frac{14035667336520444695038878437734509067558034875}{266970836062602533855168996181317985466702039577} a^{14} - \frac{40916113734303418131285937747168652940489908651}{1067883344250410135420675984725271941866808158308} a^{13} + \frac{82188004496727416117431785126853095020053200587}{533941672125205067710337992362635970933404079154} a^{12} - \frac{7104294003473358573370677098521292198889163885}{533941672125205067710337992362635970933404079154} a^{11} + \frac{41983211208630467436869841388380767706626922372}{266970836062602533855168996181317985466702039577} a^{10} - \frac{28646886753883623320494867357774999368169477887}{533941672125205067710337992362635970933404079154} a^{9} + \frac{42748269356310222861247107053278710583831577504}{266970836062602533855168996181317985466702039577} a^{8} - \frac{128315402766674301391254930576954918889366568929}{1067883344250410135420675984725271941866808158308} a^{7} - \frac{227268418592083839560034548504834627197654029187}{533941672125205067710337992362635970933404079154} a^{6} + \frac{156285399390545082984837999479828873695331009955}{1067883344250410135420675984725271941866808158308} a^{5} - \frac{4634177558493123934916743211814427489367773992}{266970836062602533855168996181317985466702039577} a^{4} - \frac{16406357609985077491639434863307866081837480597}{1067883344250410135420675984725271941866808158308} a^{3} - \frac{109596785495822897110115154314064217707095461768}{266970836062602533855168996181317985466702039577} a^{2} + \frac{51136995576321025243266361702289823130267588315}{1067883344250410135420675984725271941866808158308} a - \frac{19832017883777064939211683085463048873329931409}{533941672125205067710337992362635970933404079154}$

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:  \( 51109599709.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.170048.1, 10.6.925322313728.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{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}$
2657Data not computed