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Properties

Label 20.10.1467214799...9375.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $45.53$
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![-55, 285, 279, -2450, 6387, -5816, 165, 3648, -4889, 3043, -2239, 579, 1397, -596, 246, -253, -15, 73, -14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 14*x^18 + 73*x^17 - 15*x^16 - 253*x^15 + 246*x^14 - 596*x^13 + 1397*x^12 + 579*x^11 - 2239*x^10 + 3043*x^9 - 4889*x^8 + 3648*x^7 + 165*x^6 - 5816*x^5 + 6387*x^4 - 2450*x^3 + 279*x^2 + 285*x - 55)
 
gp: K = bnfinit(x^20 - 5*x^19 - 14*x^18 + 73*x^17 - 15*x^16 - 253*x^15 + 246*x^14 - 596*x^13 + 1397*x^12 + 579*x^11 - 2239*x^10 + 3043*x^9 - 4889*x^8 + 3648*x^7 + 165*x^6 - 5816*x^5 + 6387*x^4 - 2450*x^3 + 279*x^2 + 285*x - 55, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 14 x^{18} + 73 x^{17} - 15 x^{16} - 253 x^{15} + 246 x^{14} - 596 x^{13} + 1397 x^{12} + 579 x^{11} - 2239 x^{10} + 3043 x^{9} - 4889 x^{8} + 3648 x^{7} + 165 x^{6} - 5816 x^{5} + 6387 x^{4} - 2450 x^{3} + 279 x^{2} + 285 x - 55 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1467214799841538367562738193359375=-\,3^{10}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} + \frac{1}{4} a^{10} - \frac{3}{8} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{5104} a^{18} - \frac{53}{2552} a^{17} + \frac{89}{5104} a^{16} - \frac{111}{2552} a^{15} + \frac{3}{1276} a^{14} - \frac{543}{5104} a^{13} - \frac{691}{5104} a^{12} + \frac{51}{638} a^{11} - \frac{75}{232} a^{10} - \frac{1587}{5104} a^{9} + \frac{37}{232} a^{8} + \frac{27}{88} a^{7} + \frac{295}{5104} a^{6} - \frac{1613}{5104} a^{5} - \frac{1935}{5104} a^{4} + \frac{1229}{2552} a^{3} - \frac{353}{2552} a^{2} - \frac{599}{2552} a - \frac{47}{464}$, $\frac{1}{52018557802212855469865600245136} a^{19} - \frac{3161148124124606144231977487}{52018557802212855469865600245136} a^{18} - \frac{50795841602524126547774522347}{1793743372490098464478124146384} a^{17} + \frac{665387436726112453983977698451}{52018557802212855469865600245136} a^{16} + \frac{277688689307567250482993358706}{3251159862638303466866600015321} a^{15} + \frac{1736133892211097562043401668137}{52018557802212855469865600245136} a^{14} - \frac{6141892760558062462008604306223}{26009278901106427734932800122568} a^{13} - \frac{12114692174553630323292658851025}{52018557802212855469865600245136} a^{12} - \frac{2443402888059565186886132353741}{26009278901106427734932800122568} a^{11} - \frac{7741334713266982909683657082393}{52018557802212855469865600245136} a^{10} + \frac{7861063054837730745961492310961}{52018557802212855469865600245136} a^{9} + \frac{10894239860902667249227657376793}{26009278901106427734932800122568} a^{8} - \frac{24520076493497796533535903319873}{52018557802212855469865600245136} a^{7} - \frac{1507350400852465489018591695414}{3251159862638303466866600015321} a^{6} + \frac{411351704880923726406373320453}{896871686245049232239062073192} a^{5} + \frac{6497289364836370089030629783071}{52018557802212855469865600245136} a^{4} + \frac{1116723460100098521839211710945}{6502319725276606933733200030642} a^{3} - \frac{1983387220722488818817688473197}{26009278901106427734932800122568} a^{2} + \frac{14997637249343836224369683534031}{52018557802212855469865600245136} a + \frac{1024368219067091389882728533923}{4728959800201168679078690931376}$

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:  $14$
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:  \( 3803777303.35 \) (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 $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed