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Properties

Label 20.8.42843236564...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 97^{2}\cdot 419^{4}\cdot 695771^{2}$
Root discriminant $67.86$
Ramified primes $5, 97, 419, 695771$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94715791, -41632095, 2610644, -2328854, -47374562, 30158869, 9713821, -4911960, 1210580, -1369808, -257213, 102939, 23645, 22872, -640, -648, -29, -43, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 7*x^18 - 43*x^17 - 29*x^16 - 648*x^15 - 640*x^14 + 22872*x^13 + 23645*x^12 + 102939*x^11 - 257213*x^10 - 1369808*x^9 + 1210580*x^8 - 4911960*x^7 + 9713821*x^6 + 30158869*x^5 - 47374562*x^4 - 2328854*x^3 + 2610644*x^2 - 41632095*x + 94715791)
 
gp: K = bnfinit(x^20 - 2*x^19 - 7*x^18 - 43*x^17 - 29*x^16 - 648*x^15 - 640*x^14 + 22872*x^13 + 23645*x^12 + 102939*x^11 - 257213*x^10 - 1369808*x^9 + 1210580*x^8 - 4911960*x^7 + 9713821*x^6 + 30158869*x^5 - 47374562*x^4 - 2328854*x^3 + 2610644*x^2 - 41632095*x + 94715791, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 7 x^{18} - 43 x^{17} - 29 x^{16} - 648 x^{15} - 640 x^{14} + 22872 x^{13} + 23645 x^{12} + 102939 x^{11} - 257213 x^{10} - 1369808 x^{9} + 1210580 x^{8} - 4911960 x^{7} + 9713821 x^{6} + 30158869 x^{5} - 47374562 x^{4} - 2328854 x^{3} + 2610644 x^{2} - 41632095 x + 94715791 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4284323656481291490575776641845703125=5^{15}\cdot 97^{2}\cdot 419^{4}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.86$
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:  $5, 97, 419, 695771$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{19} + \frac{1976976090175849095029954243060453705652365775132471375513558739340611716227371771988}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{18} - \frac{265250096692728398984985167451474947078173992561995726175901630688867113306946347275}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{17} - \frac{970439085227012415768625345754543965256344675321316297977012459092510253321628944392}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{16} - \frac{2288257768852113404925318548338601876714439545703644577958138644755398658067852510515}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{15} + \frac{882499177249471712811114933632364849519309034111971148214974336399826442597178009287}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{14} + \frac{1692864223719866114201202018423678506532193795268149603871297483402897822109863475210}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{13} - \frac{887907775066062169174185009258752711246818282589661975215281727219268150824269006742}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{12} + \frac{288637860002134726336814247840319314642029538271607878896126398666963189996809458556}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{11} - \frac{2755913155155992007281081893387236535059976099575374460963657400460650345740822113504}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{10} + \frac{599928911310311749753994466579388770276043818247477454184778871568042166091789813859}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{9} + \frac{78974198599633782484768540541658844172429971838275513752014246465417522597799993326}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{8} + \frac{2382759826311382176272638797837414711020324306285125638182207018967054579961464960614}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{7} + \frac{411870153387641367002763718177921151852385254565548779072664249088103530329249969409}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{6} - \frac{960460899124337710175297682062008957725528070695168097860336495213577531699605736621}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{5} + \frac{2404072978604085917925617917626638473210490450814376202664037892495528074763519740964}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{4} + \frac{2487464540053611692394151342956123519601873586104908419921196703019720373974985853672}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{3} - \frac{1017573431643372944271539032497793864409313237783264185472972985838038566359447595551}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a^{2} - \frac{2658303121195356743662532326875478353659489985846127881265953328245311240530955944194}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471} a + \frac{413306853588332665143236177095111218570527504211721436892784485247646140716478877680}{5529185545630690597491968386628758228428068351153578010548702484750433873484417879471}$

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:  $13$
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:  \( 23510348708.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
419Data not computed
695771Data not computed