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Properties

Label 20.12.3879990017...9609.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{2}\cdot 401^{11}$
Root discriminant $30.16$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288, 2784, -3848, -12852, 22918, 5789, -35470, 23309, 7037, -19450, 13038, -2829, -2597, 2973, -1411, 240, 127, -114, 45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 114*x^17 + 127*x^16 + 240*x^15 - 1411*x^14 + 2973*x^13 - 2597*x^12 - 2829*x^11 + 13038*x^10 - 19450*x^9 + 7037*x^8 + 23309*x^7 - 35470*x^6 + 5789*x^5 + 22918*x^4 - 12852*x^3 - 3848*x^2 + 2784*x + 288)
 
gp: K = bnfinit(x^20 - 10*x^19 + 45*x^18 - 114*x^17 + 127*x^16 + 240*x^15 - 1411*x^14 + 2973*x^13 - 2597*x^12 - 2829*x^11 + 13038*x^10 - 19450*x^9 + 7037*x^8 + 23309*x^7 - 35470*x^6 + 5789*x^5 + 22918*x^4 - 12852*x^3 - 3848*x^2 + 2784*x + 288, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 45 x^{18} - 114 x^{17} + 127 x^{16} + 240 x^{15} - 1411 x^{14} + 2973 x^{13} - 2597 x^{12} - 2829 x^{11} + 13038 x^{10} - 19450 x^{9} + 7037 x^{8} + 23309 x^{7} - 35470 x^{6} + 5789 x^{5} + 22918 x^{4} - 12852 x^{3} - 3848 x^{2} + 2784 x + 288 \)

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:  \(387999001772721822047119239609=3^{2}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.16$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{24} a^{16} + \frac{1}{12} a^{15} + \frac{1}{12} a^{14} - \frac{1}{6} a^{13} + \frac{1}{24} a^{12} - \frac{1}{4} a^{10} + \frac{1}{24} a^{9} + \frac{1}{24} a^{8} - \frac{1}{6} a^{7} - \frac{3}{8} a^{6} + \frac{5}{12} a^{5} - \frac{1}{3} a^{4} - \frac{5}{24} a^{3} - \frac{5}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{24} a^{17} - \frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{1}{8} a^{13} + \frac{1}{6} a^{12} + \frac{1}{24} a^{10} + \frac{11}{24} a^{9} - \frac{1}{4} a^{8} + \frac{5}{24} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{7}{24} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{48} a^{18} - \frac{1}{48} a^{17} - \frac{1}{24} a^{15} - \frac{1}{16} a^{14} + \frac{5}{48} a^{13} - \frac{1}{24} a^{12} - \frac{5}{48} a^{11} + \frac{11}{24} a^{10} - \frac{1}{16} a^{9} + \frac{7}{48} a^{8} + \frac{7}{16} a^{7} + \frac{11}{24} a^{6} + \frac{11}{48} a^{5} - \frac{1}{16} a^{4} + \frac{1}{8} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{2930546426972094741072} a^{19} - \frac{9889040452597222837}{976848808990698247024} a^{18} + \frac{16462561183242982}{1711767772764074031} a^{17} - \frac{20812119493508991409}{1465273213486047370536} a^{16} + \frac{72966298576453995217}{2930546426972094741072} a^{15} - \frac{491732174634027901}{2930546426972094741072} a^{14} + \frac{14205154657089663133}{488424404495349123512} a^{13} - \frac{2514940038278406697}{2930546426972094741072} a^{12} + \frac{8721374780536243013}{183159151685755921317} a^{11} - \frac{12546675337742943725}{2930546426972094741072} a^{10} - \frac{1347413068853899159169}{2930546426972094741072} a^{9} - \frac{1168586217444879221213}{2930546426972094741072} a^{8} - \frac{76179971233567537069}{1465273213486047370536} a^{7} + \frac{17688298409167324965}{976848808990698247024} a^{6} + \frac{787951979994733289735}{2930546426972094741072} a^{5} - \frac{202437235842516759209}{488424404495349123512} a^{4} + \frac{4439557376929600529}{23633438927194312428} a^{3} + \frac{133496835320496052559}{366318303371511842634} a^{2} - \frac{4818000724839109337}{11816719463597156214} a - \frac{1480065002976305257}{61053050561918640439}$

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:  \( 51324957.9239 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed