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Properties

Label 20.14.4075596666...9375.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,3^{8}\cdot 5^{12}\cdot 239^{9}$
Root discriminant $47.92$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4491, -3750, 26654, 13443, -30395, -9863, -32803, -23680, 66757, 34151, -33252, -13918, 5860, 1548, -344, 109, 58, -19, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 - 19*x^17 + 58*x^16 + 109*x^15 - 344*x^14 + 1548*x^13 + 5860*x^12 - 13918*x^11 - 33252*x^10 + 34151*x^9 + 66757*x^8 - 23680*x^7 - 32803*x^6 - 9863*x^5 - 30395*x^4 + 13443*x^3 + 26654*x^2 - 3750*x - 4491)
 
gp: K = bnfinit(x^20 - 15*x^18 - 19*x^17 + 58*x^16 + 109*x^15 - 344*x^14 + 1548*x^13 + 5860*x^12 - 13918*x^11 - 33252*x^10 + 34151*x^9 + 66757*x^8 - 23680*x^7 - 32803*x^6 - 9863*x^5 - 30395*x^4 + 13443*x^3 + 26654*x^2 - 3750*x - 4491, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} - 19 x^{17} + 58 x^{16} + 109 x^{15} - 344 x^{14} + 1548 x^{13} + 5860 x^{12} - 13918 x^{11} - 33252 x^{10} + 34151 x^{9} + 66757 x^{8} - 23680 x^{7} - 32803 x^{6} - 9863 x^{5} - 30395 x^{4} + 13443 x^{3} + 26654 x^{2} - 3750 x - 4491 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4075596666226495465452050537109375=-\,3^{8}\cdot 5^{12}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.92$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{17} - \frac{2}{9} a^{16} + \frac{1}{9} a^{15} + \frac{4}{9} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{4}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{6} - \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9598746439564483564928066406945878155705281477} a^{19} - \frac{384688223546576264070701577269228922771613297}{9598746439564483564928066406945878155705281477} a^{18} - \frac{386137689510169015235224738585214284829203640}{9598746439564483564928066406945878155705281477} a^{17} + \frac{121072446804478049316325921007953202627770577}{417336801720194937605568104649820789378490499} a^{16} + \frac{155611373617174996270536706378465364490922765}{1066527382173831507214229600771764239522809053} a^{15} - \frac{3309276169166422250342945133520178992506821162}{9598746439564483564928066406945878155705281477} a^{14} + \frac{1391076093314718358319009245334937882088031398}{3199582146521494521642688802315292718568427159} a^{13} + \frac{25403120774840612135351295922458529928032461}{96957034743075591564929963706524021774800823} a^{12} - \frac{68028609455154761739587851315888338772749482}{505197181029709661312003495102414639773962183} a^{11} - \frac{4465616181446044696411032401920132492345583594}{9598746439564483564928066406945878155705281477} a^{10} + \frac{1775385371533984878330775339463590431541215152}{9598746439564483564928066406945878155705281477} a^{9} - \frac{48985333737325591054689145041101355160767083}{3199582146521494521642688802315292718568427159} a^{8} + \frac{9215779239114068214855087771019475991335585}{872613312687680324084369673358716195973207407} a^{7} + \frac{969046195887356159559485472946272257148342385}{9598746439564483564928066406945878155705281477} a^{6} - \frac{753243169672370743771353844211528282632860011}{9598746439564483564928066406945878155705281477} a^{5} - \frac{119939163198393656512744659384385834300260808}{1066527382173831507214229600771764239522809053} a^{4} + \frac{2651407672089722620181874265139064122917166350}{9598746439564483564928066406945878155705281477} a^{3} - \frac{1005104034018813900824026681558691848628231941}{9598746439564483564928066406945878155705281477} a^{2} - \frac{1248720669820237384808255064698274080466350114}{3199582146521494521642688802315292718568427159} a + \frac{399821602919679167099529762066458962874051715}{1066527382173831507214229600771764239522809053}$

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:  $16$
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:  \( 13228720565.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$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}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed