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Properties

Label 20.8.12154762819...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{16}\cdot 6029^{5}$
Root discriminant $31.93$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![251, -215, -733, -551, -970, 4645, 3448, -6778, -428, 3671, -2827, -431, 1350, -581, -22, 205, -77, 11, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 + 11*x^17 - 77*x^16 + 205*x^15 - 22*x^14 - 581*x^13 + 1350*x^12 - 431*x^11 - 2827*x^10 + 3671*x^9 - 428*x^8 - 6778*x^7 + 3448*x^6 + 4645*x^5 - 970*x^4 - 551*x^3 - 733*x^2 - 215*x + 251)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 + 11*x^17 - 77*x^16 + 205*x^15 - 22*x^14 - 581*x^13 + 1350*x^12 - 431*x^11 - 2827*x^10 + 3671*x^9 - 428*x^8 - 6778*x^7 + 3448*x^6 + 4645*x^5 - 970*x^4 - 551*x^3 - 733*x^2 - 215*x + 251, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} + 11 x^{17} - 77 x^{16} + 205 x^{15} - 22 x^{14} - 581 x^{13} + 1350 x^{12} - 431 x^{11} - 2827 x^{10} + 3671 x^{9} - 428 x^{8} - 6778 x^{7} + 3448 x^{6} + 4645 x^{5} - 970 x^{4} - 551 x^{3} - 733 x^{2} - 215 x + 251 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1215476281933432182159423828125=5^{16}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.93$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{13} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{1175} a^{18} - \frac{72}{1175} a^{17} + \frac{18}{235} a^{16} - \frac{107}{1175} a^{15} + \frac{49}{1175} a^{14} + \frac{24}{235} a^{13} + \frac{469}{1175} a^{12} - \frac{98}{1175} a^{11} - \frac{19}{235} a^{10} + \frac{242}{1175} a^{9} + \frac{432}{1175} a^{8} - \frac{432}{1175} a^{7} + \frac{221}{1175} a^{6} + \frac{106}{1175} a^{5} - \frac{256}{1175} a^{4} + \frac{357}{1175} a^{3} + \frac{44}{235} a^{2} - \frac{53}{1175} a + \frac{381}{1175}$, $\frac{1}{12418814495352483008464843475} a^{19} + \frac{191376880555479529618835}{496752579814099320338593739} a^{18} - \frac{87315690184297355388734049}{12418814495352483008464843475} a^{17} + \frac{98919141633768870015142943}{12418814495352483008464843475} a^{16} + \frac{66801328367235263712394151}{2483762899070496601692968695} a^{15} + \frac{884768244971315427816010408}{12418814495352483008464843475} a^{14} + \frac{10208004049027282311842132}{264230095645797510818400925} a^{13} + \frac{339310481027127168215185207}{2483762899070496601692968695} a^{12} + \frac{5011961757180034730342369724}{12418814495352483008464843475} a^{11} + \frac{1212043573195240391595516622}{12418814495352483008464843475} a^{10} + \frac{162792744109556976638829476}{12418814495352483008464843475} a^{9} - \frac{4682282432801652273065256943}{12418814495352483008464843475} a^{8} + \frac{1494327478403060357918552432}{12418814495352483008464843475} a^{7} + \frac{3955056083762871034871034118}{12418814495352483008464843475} a^{6} - \frac{1785662608324150562381430099}{12418814495352483008464843475} a^{5} - \frac{1088095426978266281483341986}{2483762899070496601692968695} a^{4} - \frac{2307513019241699876100134406}{12418814495352483008464843475} a^{3} - \frac{4589966616975032304817735788}{12418814495352483008464843475} a^{2} - \frac{752339798638950197557777126}{2483762899070496601692968695} a - \frac{2836377848428843731074780423}{12418814495352483008464843475}$

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:  $13$
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:  \( 25262835.879 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed