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Properties

Label 20.10.8151193332...1875.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{8}\cdot 5^{11}\cdot 239^{9}$
Root discriminant $44.21$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![605, 1605, -5890, -2255, -16959, 33765, 47350, -54399, 12614, -11008, 2973, -894, 3422, -489, -539, 148, -93, 45, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 3*x^18 + 45*x^17 - 93*x^16 + 148*x^15 - 539*x^14 - 489*x^13 + 3422*x^12 - 894*x^11 + 2973*x^10 - 11008*x^9 + 12614*x^8 - 54399*x^7 + 47350*x^6 + 33765*x^5 - 16959*x^4 - 2255*x^3 - 5890*x^2 + 1605*x + 605)
 
gp: K = bnfinit(x^20 - 6*x^19 + 3*x^18 + 45*x^17 - 93*x^16 + 148*x^15 - 539*x^14 - 489*x^13 + 3422*x^12 - 894*x^11 + 2973*x^10 - 11008*x^9 + 12614*x^8 - 54399*x^7 + 47350*x^6 + 33765*x^5 - 16959*x^4 - 2255*x^3 - 5890*x^2 + 1605*x + 605, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 3 x^{18} + 45 x^{17} - 93 x^{16} + 148 x^{15} - 539 x^{14} - 489 x^{13} + 3422 x^{12} - 894 x^{11} + 2973 x^{10} - 11008 x^{9} + 12614 x^{8} - 54399 x^{7} + 47350 x^{6} + 33765 x^{5} - 16959 x^{4} - 2255 x^{3} - 5890 x^{2} + 1605 x + 605 \)

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:  \(-815119333245299093090410107421875=-\,3^{8}\cdot 5^{11}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.21$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{45} a^{16} + \frac{1}{15} a^{15} + \frac{7}{45} a^{14} - \frac{1}{15} a^{13} - \frac{2}{15} a^{12} + \frac{7}{45} a^{11} + \frac{4}{45} a^{9} - \frac{22}{45} a^{8} - \frac{4}{15} a^{7} + \frac{8}{45} a^{6} + \frac{13}{45} a^{5} + \frac{14}{45} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{45} a^{17} - \frac{2}{45} a^{15} + \frac{2}{15} a^{14} + \frac{1}{15} a^{13} - \frac{1}{9} a^{12} - \frac{2}{15} a^{11} + \frac{4}{45} a^{10} - \frac{4}{45} a^{9} - \frac{7}{15} a^{8} - \frac{16}{45} a^{7} + \frac{19}{45} a^{6} + \frac{4}{9} a^{5} - \frac{2}{45} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a$, $\frac{1}{135} a^{18} - \frac{1}{135} a^{16} - \frac{2}{45} a^{15} + \frac{2}{27} a^{14} + \frac{7}{135} a^{13} + \frac{1}{45} a^{12} - \frac{4}{135} a^{11} + \frac{11}{135} a^{10} - \frac{2}{135} a^{9} - \frac{53}{135} a^{8} - \frac{23}{135} a^{7} - \frac{47}{135} a^{6} + \frac{41}{135} a^{5} - \frac{2}{45} a^{4} - \frac{1}{27} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{10}{27}$, $\frac{1}{465738441781175914923822059182912059231075} a^{19} - \frac{327910991587015092420763600085998545026}{155246147260391971641274019727637353077025} a^{18} + \frac{326003718245326290117755772187694645459}{465738441781175914923822059182912059231075} a^{17} - \frac{114661403961265011215940579978319171807}{51748715753463990547091339909212451025675} a^{16} - \frac{67936696259343471054166606281528607213367}{465738441781175914923822059182912059231075} a^{15} + \frac{49237103459852472503996732424466424038942}{465738441781175914923822059182912059231075} a^{14} - \frac{17210051219851382798461983138631419480646}{155246147260391971641274019727637353077025} a^{13} + \frac{64823564166109307722929821780479391438732}{465738441781175914923822059182912059231075} a^{12} + \frac{42086650494757186422226892171160543618008}{465738441781175914923822059182912059231075} a^{11} - \frac{3744640633205766295097646596996797428232}{93147688356235182984764411836582411846215} a^{10} - \frac{190689371338653070174734308466962864168807}{465738441781175914923822059182912059231075} a^{9} + \frac{185435259434077503319798798234373135061076}{465738441781175914923822059182912059231075} a^{8} - \frac{45954938297839522043862449451464855292578}{465738441781175914923822059182912059231075} a^{7} - \frac{51369661964501139109107262994405933364403}{465738441781175914923822059182912059231075} a^{6} - \frac{66872567306294839180317361163887681365913}{155246147260391971641274019727637353077025} a^{5} + \frac{206720356116400203410748353608930249048078}{465738441781175914923822059182912059231075} a^{4} + \frac{1183757275176051325477168217625878632949}{6209845890415678865650960789105494123081} a^{3} + \frac{2594669145304868875234764782255018811328}{31049229452078394328254803945527470615405} a^{2} + \frac{23589803441067076046599050663112114642864}{93147688356235182984764411836582411846215} a - \frac{5849595204887861491446244301381797017029}{31049229452078394328254803945527470615405}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1310163116.31 \) (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 $20$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$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}$
239Data not computed