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Properties

Label 20.12.5258696461...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{15}\cdot 11^{6}\cdot 9931^{4}$
Root discriminant $43.26$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50051, -33500, -231764, 183454, 387768, -352758, -288014, 301874, 104483, -130158, -20568, 30457, 2095, -3221, -195, -98, 61, 62, -15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 15*x^18 + 62*x^17 + 61*x^16 - 98*x^15 - 195*x^14 - 3221*x^13 + 2095*x^12 + 30457*x^11 - 20568*x^10 - 130158*x^9 + 104483*x^8 + 301874*x^7 - 288014*x^6 - 352758*x^5 + 387768*x^4 + 183454*x^3 - 231764*x^2 - 33500*x + 50051)
 
gp: K = bnfinit(x^20 - 4*x^19 - 15*x^18 + 62*x^17 + 61*x^16 - 98*x^15 - 195*x^14 - 3221*x^13 + 2095*x^12 + 30457*x^11 - 20568*x^10 - 130158*x^9 + 104483*x^8 + 301874*x^7 - 288014*x^6 - 352758*x^5 + 387768*x^4 + 183454*x^3 - 231764*x^2 - 33500*x + 50051, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 15 x^{18} + 62 x^{17} + 61 x^{16} - 98 x^{15} - 195 x^{14} - 3221 x^{13} + 2095 x^{12} + 30457 x^{11} - 20568 x^{10} - 130158 x^{9} + 104483 x^{8} + 301874 x^{7} - 288014 x^{6} - 352758 x^{5} + 387768 x^{4} + 183454 x^{3} - 231764 x^{2} - 33500 x + 50051 \)

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:  \(525869646190169848201934814453125=5^{15}\cdot 11^{6}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.26$
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:  $5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1497988253937515399353683964446095417} a^{19} + \frac{467855462367323329254451313031340894}{1497988253937515399353683964446095417} a^{18} + \frac{325486504869980026861991017521961501}{1497988253937515399353683964446095417} a^{17} + \frac{307749432838398482762595474389995905}{1497988253937515399353683964446095417} a^{16} - \frac{369433243449957403411062375886764094}{1497988253937515399353683964446095417} a^{15} + \frac{54686584039681593739070034464271089}{1497988253937515399353683964446095417} a^{14} - \frac{130463430171032029726538327660607074}{1497988253937515399353683964446095417} a^{13} + \frac{260806490366301366721066824426390122}{1497988253937515399353683964446095417} a^{12} - \frac{44034363328772717648963810293738632}{1497988253937515399353683964446095417} a^{11} + \frac{44695768033237521571122295595411416}{1497988253937515399353683964446095417} a^{10} + \frac{652889594553840715642908582006034026}{1497988253937515399353683964446095417} a^{9} - \frac{524480090651952799345791351715198238}{1497988253937515399353683964446095417} a^{8} - \frac{627121269617049040418843527642846024}{1497988253937515399353683964446095417} a^{7} - \frac{441945169727228404305997052041174009}{1497988253937515399353683964446095417} a^{6} + \frac{200035067114697044570808607989925957}{1497988253937515399353683964446095417} a^{5} - \frac{693227514371108491220900491017672211}{1497988253937515399353683964446095417} a^{4} + \frac{237830794391087219823809263601085592}{1497988253937515399353683964446095417} a^{3} - \frac{497029319379127330242272585135906628}{1497988253937515399353683964446095417} a^{2} - \frac{57477755415741131486355238029634516}{1497988253937515399353683964446095417} a + \frac{536699190528696116876449956944656246}{1497988253937515399353683964446095417}$

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:  \( 2510789862.49 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

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 $20$ R $20$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
9931Data not computed