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Properties

Label 20.0.17672461644...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{14}\cdot 6029^{7}$
Root discriminant $64.92$
Ramified primes $5, 6029$
Class number $16350$ (GRH)
Class group $[16350]$ (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11369359, -43507098, 98000344, -134835408, 155067625, -130092378, 97213725, -57126905, 30194239, -13137281, 5124144, -1709342, 515016, -137685, 33532, -7304, 1502, -231, 49, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 49*x^18 - 231*x^17 + 1502*x^16 - 7304*x^15 + 33532*x^14 - 137685*x^13 + 515016*x^12 - 1709342*x^11 + 5124144*x^10 - 13137281*x^9 + 30194239*x^8 - 57126905*x^7 + 97213725*x^6 - 130092378*x^5 + 155067625*x^4 - 134835408*x^3 + 98000344*x^2 - 43507098*x + 11369359)
 
gp: K = bnfinit(x^20 - 3*x^19 + 49*x^18 - 231*x^17 + 1502*x^16 - 7304*x^15 + 33532*x^14 - 137685*x^13 + 515016*x^12 - 1709342*x^11 + 5124144*x^10 - 13137281*x^9 + 30194239*x^8 - 57126905*x^7 + 97213725*x^6 - 130092378*x^5 + 155067625*x^4 - 134835408*x^3 + 98000344*x^2 - 43507098*x + 11369359, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 49 x^{18} - 231 x^{17} + 1502 x^{16} - 7304 x^{15} + 33532 x^{14} - 137685 x^{13} + 515016 x^{12} - 1709342 x^{11} + 5124144 x^{10} - 13137281 x^{9} + 30194239 x^{8} - 57126905 x^{7} + 97213725 x^{6} - 130092378 x^{5} + 155067625 x^{4} - 134835408 x^{3} + 98000344 x^{2} - 43507098 x + 11369359 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1767246164450779958943837335205078125=5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.92$
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:  $5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{3275} a^{18} + \frac{133}{3275} a^{17} - \frac{182}{3275} a^{16} + \frac{16}{655} a^{15} + \frac{9}{655} a^{14} - \frac{159}{3275} a^{13} - \frac{247}{3275} a^{12} + \frac{924}{3275} a^{11} + \frac{1453}{3275} a^{10} + \frac{214}{655} a^{9} - \frac{1393}{3275} a^{8} + \frac{1411}{3275} a^{7} + \frac{907}{3275} a^{6} - \frac{712}{3275} a^{5} + \frac{19}{655} a^{4} - \frac{186}{655} a^{3} + \frac{316}{655} a^{2} + \frac{1087}{3275} a - \frac{9}{25}$, $\frac{1}{24911587955422366171776634872407139525395616268283038859663240413525} a^{19} + \frac{1542505497553985144462622983451704025012635899824505720536843699}{24911587955422366171776634872407139525395616268283038859663240413525} a^{18} - \frac{1810773649957383529493602206815355209821726797683855647960624580454}{24911587955422366171776634872407139525395616268283038859663240413525} a^{17} + \frac{1221098071458804044717692212316423617102448575542187590062535872053}{24911587955422366171776634872407139525395616268283038859663240413525} a^{16} + \frac{327065924381160022421453190288226132807118725292229344703078089836}{4982317591084473234355326974481427905079123253656607771932648082705} a^{15} - \frac{270441574109249379710645083649555851779218911531032009566778307634}{24911587955422366171776634872407139525395616268283038859663240413525} a^{14} - \frac{680031543158912241506200057128348644723463154679309446906633689186}{24911587955422366171776634872407139525395616268283038859663240413525} a^{13} + \frac{1892570216225947782600829466265458670970085134920543192347778392452}{24911587955422366171776634872407139525395616268283038859663240413525} a^{12} + \frac{9771810793200160822972615070655960391521007888939273094158806605572}{24911587955422366171776634872407139525395616268283038859663240413525} a^{11} + \frac{6117256726289279551314259805831805988432915164768535523411940840263}{24911587955422366171776634872407139525395616268283038859663240413525} a^{10} + \frac{4854026798817779186791065189003343311722841592312052765917136329642}{24911587955422366171776634872407139525395616268283038859663240413525} a^{9} + \frac{758642286933823807229228591216997942647974386156587738720728964158}{24911587955422366171776634872407139525395616268283038859663240413525} a^{8} + \frac{58729323075182893260135144524548887724386279283998433455379089268}{24911587955422366171776634872407139525395616268283038859663240413525} a^{7} - \frac{331281563789295592353164215785245390566600991733446576247812346638}{996463518216894646871065394896285581015824650731321554386529616541} a^{6} - \frac{4722307911273592342386371534274865756372943592018670971949397750192}{24911587955422366171776634872407139525395616268283038859663240413525} a^{5} - \frac{2050989570798785448722022918959516758889129318817817333136086901918}{4982317591084473234355326974481427905079123253656607771932648082705} a^{4} - \frac{323006687047485344559640148378608998665388487628965256345801536485}{996463518216894646871065394896285581015824650731321554386529616541} a^{3} + \frac{3229037207794615735340740083051979846796693802177953740457196516922}{24911587955422366171776634872407139525395616268283038859663240413525} a^{2} + \frac{4002973773514734579785070854103626773349492463968074529297227479843}{24911587955422366171776634872407139525395616268283038859663240413525} a - \frac{33556135266592313206713280948871988399543209069966207405831657624}{190164793552842489860890342537459080346531421895290372974528552775}$

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

Class group and class number

$C_{16350}$, which has order $16350$ (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:  $9$
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:  \( 341439.528105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.150725.1, 5.5.753625.1, 10.10.2839753203125.1

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 24 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$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}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed