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Properties

Label 20.10.4075596666...9375.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,3^{8}\cdot 5^{12}\cdot 239^{9}$
Root discriminant $47.92$
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![-9871, 57792, -118335, 101145, 10046, -123767, 139583, -52485, -41659, 66812, -36253, 2065, 9906, -6707, 1670, 330, -397, 127, -9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 9*x^18 + 127*x^17 - 397*x^16 + 330*x^15 + 1670*x^14 - 6707*x^13 + 9906*x^12 + 2065*x^11 - 36253*x^10 + 66812*x^9 - 41659*x^8 - 52485*x^7 + 139583*x^6 - 123767*x^5 + 10046*x^4 + 101145*x^3 - 118335*x^2 + 57792*x - 9871)
 
gp: K = bnfinit(x^20 - 5*x^19 - 9*x^18 + 127*x^17 - 397*x^16 + 330*x^15 + 1670*x^14 - 6707*x^13 + 9906*x^12 + 2065*x^11 - 36253*x^10 + 66812*x^9 - 41659*x^8 - 52485*x^7 + 139583*x^6 - 123767*x^5 + 10046*x^4 + 101145*x^3 - 118335*x^2 + 57792*x - 9871, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 9 x^{18} + 127 x^{17} - 397 x^{16} + 330 x^{15} + 1670 x^{14} - 6707 x^{13} + 9906 x^{12} + 2065 x^{11} - 36253 x^{10} + 66812 x^{9} - 41659 x^{8} - 52485 x^{7} + 139583 x^{6} - 123767 x^{5} + 10046 x^{4} + 101145 x^{3} - 118335 x^{2} + 57792 x - 9871 \)

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:  \(-4075596666226495465452050537109375=-\,3^{8}\cdot 5^{12}\cdot 239^{9}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{45} a^{18} - \frac{7}{45} a^{17} - \frac{2}{15} a^{16} + \frac{2}{15} a^{15} + \frac{2}{45} a^{14} + \frac{1}{9} a^{13} + \frac{2}{5} a^{12} - \frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{2}{9} a^{9} - \frac{4}{15} a^{8} + \frac{7}{15} a^{7} - \frac{19}{45} a^{6} + \frac{2}{45} a^{5} + \frac{2}{5} a^{4} - \frac{1}{3} a^{3} - \frac{7}{45} a^{2} - \frac{16}{45} a + \frac{4}{45}$, $\frac{1}{489558835846577725198358569752494107125} a^{19} - \frac{57685293096840491168934400508880622}{21285166775938161965146024771847569875} a^{18} + \frac{55663190861929408681920009459196593097}{489558835846577725198358569752494107125} a^{17} + \frac{1249378232563655288491859107212783029}{10879085241035060559963523772277646825} a^{16} + \frac{62340934856133216222521202785105005298}{489558835846577725198358569752494107125} a^{15} + \frac{5583590641375341887289500850638823348}{54395426205175302799817618861388234125} a^{14} - \frac{128479462210683455892677223863555183687}{489558835846577725198358569752494107125} a^{13} + \frac{9198237599699839486491654548636711497}{32637255723105181679890571316832940475} a^{12} + \frac{28192988916657049496162284464959414692}{163186278615525908399452856584164702375} a^{11} + \frac{54367089380915646570005065026614524939}{489558835846577725198358569752494107125} a^{10} - \frac{127076198643352455785579978185252317892}{489558835846577725198358569752494107125} a^{9} - \frac{38313518956560858656074269084192857}{163186278615525908399452856584164702375} a^{8} + \frac{63659174373759773026434226575576529337}{489558835846577725198358569752494107125} a^{7} - \frac{44499615435452785702983799438592147099}{163186278615525908399452856584164702375} a^{6} - \frac{20367597521393037725258527359532170004}{97911767169315545039671713950498821425} a^{5} + \frac{73793367876047375021141401535761776376}{163186278615525908399452856584164702375} a^{4} - \frac{154024531877569649547924331084863379357}{489558835846577725198358569752494107125} a^{3} - \frac{14646313255035728616630000054217129091}{163186278615525908399452856584164702375} a^{2} - \frac{209846400163946011480664202854578197037}{489558835846577725198358569752494107125} a - \frac{114556296799521056490517343990519420071}{489558835846577725198358569752494107125}$

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:  \( 3020014559.57 \) (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 $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
239Data not computed