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Properties

Label 20.4.61257625575...0496.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{48}\cdot 31^{10}\cdot 227^{4}$
Root discriminant $86.97$
Ramified primes $2, 31, 227$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3959762, -68625992, -67507864, 23998572, 11339568, 25190444, -4021410, -4918480, 766294, -274808, -180762, 110152, -50715, 1724, 7116, -1196, 1221, -124, 54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 54*x^18 - 124*x^17 + 1221*x^16 - 1196*x^15 + 7116*x^14 + 1724*x^13 - 50715*x^12 + 110152*x^11 - 180762*x^10 - 274808*x^9 + 766294*x^8 - 4918480*x^7 - 4021410*x^6 + 25190444*x^5 + 11339568*x^4 + 23998572*x^3 - 67507864*x^2 - 68625992*x + 3959762)
 
gp: K = bnfinit(x^20 - 4*x^19 + 54*x^18 - 124*x^17 + 1221*x^16 - 1196*x^15 + 7116*x^14 + 1724*x^13 - 50715*x^12 + 110152*x^11 - 180762*x^10 - 274808*x^9 + 766294*x^8 - 4918480*x^7 - 4021410*x^6 + 25190444*x^5 + 11339568*x^4 + 23998572*x^3 - 67507864*x^2 - 68625992*x + 3959762, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 54 x^{18} - 124 x^{17} + 1221 x^{16} - 1196 x^{15} + 7116 x^{14} + 1724 x^{13} - 50715 x^{12} + 110152 x^{11} - 180762 x^{10} - 274808 x^{9} + 766294 x^{8} - 4918480 x^{7} - 4021410 x^{6} + 25190444 x^{5} + 11339568 x^{4} + 23998572 x^{3} - 67507864 x^{2} - 68625992 x + 3959762 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(612576255759573370734751264818904170496=2^{48}\cdot 31^{10}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{300476283061578437495976186211179999393019435583366152836919512434635617122943649772} a^{19} + \frac{16871071251512066225780877857825863330860443897202426483224629171119940456674444579}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{18} + \frac{19165523025555164529460231952391907654752364176853573049257588220773996316115636883}{300476283061578437495976186211179999393019435583366152836919512434635617122943649772} a^{17} - \frac{70824644936353160677461913752133581791218770040306394665220490270467075773002733473}{300476283061578437495976186211179999393019435583366152836919512434635617122943649772} a^{16} + \frac{31297300993113511303463094675297465799007276683184561513048071242534720112300985293}{300476283061578437495976186211179999393019435583366152836919512434635617122943649772} a^{15} + \frac{59639095491101205550508760199243033753834637076090470461231938940285841704013859603}{300476283061578437495976186211179999393019435583366152836919512434635617122943649772} a^{14} - \frac{17269363737843734845014123354370347152260153665271704203102099630371594660301576705}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{13} + \frac{111047630277221029165132981787055841767889135694137702257216967439323918228625250479}{300476283061578437495976186211179999393019435583366152836919512434635617122943649772} a^{12} + \frac{47740522167981813630783523244238126467834096622146876923482475758789798048646396313}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{11} + \frac{30461050737478016195325826305659383621727711190940215321304850796392242986334824897}{75119070765394609373994046552794999848254858895841538209229878108658904280735912443} a^{10} + \frac{905379644950011655685992284752409476022117713675313777186787434869444792295823691}{75119070765394609373994046552794999848254858895841538209229878108658904280735912443} a^{9} - \frac{64110403895748092203227084396287386224600194284839535992169279976492563968596494887}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{8} + \frac{27913370438041798275951660680226789006618348331308067741004689971228137017212123697}{75119070765394609373994046552794999848254858895841538209229878108658904280735912443} a^{7} + \frac{26951323686840083348718637714199502843104238677986700692879530232646155659797031050}{75119070765394609373994046552794999848254858895841538209229878108658904280735912443} a^{6} + \frac{19924412085156180463789239550589799382393229238878337338791797497965247745999848475}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{5} - \frac{1892640436323381689502626051234757510296261695651079672927535132520933224911930973}{75119070765394609373994046552794999848254858895841538209229878108658904280735912443} a^{4} - \frac{36781567306589780756469681577508463003471423779498923353942415159319308800148316643}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{3} + \frac{37532488090537595264322187173180371335350352814763112997078893559783104414095628183}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886} a^{2} - \frac{24306140733449728566399164889973968938211178601330434361900491189649413000198181566}{75119070765394609373994046552794999848254858895841538209229878108658904280735912443} a - \frac{24264364917410867770473508275019224341780786056030239787833355264148797241278296329}{150238141530789218747988093105589999696509717791683076418459756217317808561471824886}$

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:  $11$
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:  \( 268414133640 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 R $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.10.5.1$x^{10} - 1922 x^{6} + 923521 x^{2} - 2862915100$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
227Data not computed