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Properties

Label 20.0.90998164050...3104.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 97^{2}\cdot 2657^{6}$
Root discriminant $62.80$
Ramified primes $2, 97, 2657$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![102461764, -30312416, 131507456, -46490160, 71119940, -27420408, 26644588, -10134664, 6856148, -2773796, 1294936, -528056, 179453, -70552, 19584, -5778, 1606, -292, 69, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 69*x^18 - 292*x^17 + 1606*x^16 - 5778*x^15 + 19584*x^14 - 70552*x^13 + 179453*x^12 - 528056*x^11 + 1294936*x^10 - 2773796*x^9 + 6856148*x^8 - 10134664*x^7 + 26644588*x^6 - 27420408*x^5 + 71119940*x^4 - 46490160*x^3 + 131507456*x^2 - 30312416*x + 102461764)
 
gp: K = bnfinit(x^20 - 6*x^19 + 69*x^18 - 292*x^17 + 1606*x^16 - 5778*x^15 + 19584*x^14 - 70552*x^13 + 179453*x^12 - 528056*x^11 + 1294936*x^10 - 2773796*x^9 + 6856148*x^8 - 10134664*x^7 + 26644588*x^6 - 27420408*x^5 + 71119940*x^4 - 46490160*x^3 + 131507456*x^2 - 30312416*x + 102461764, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 69 x^{18} - 292 x^{17} + 1606 x^{16} - 5778 x^{15} + 19584 x^{14} - 70552 x^{13} + 179453 x^{12} - 528056 x^{11} + 1294936 x^{10} - 2773796 x^{9} + 6856148 x^{8} - 10134664 x^{7} + 26644588 x^{6} - 27420408 x^{5} + 71119940 x^{4} - 46490160 x^{3} + 131507456 x^{2} - 30312416 x + 102461764 \)

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:  \(909981640506746603991752188943663104=2^{38}\cdot 97^{2}\cdot 2657^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.80$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{19} - \frac{5922219330277504319431053204807879994237345162499843076039177936617164353}{115967925536511885560973953639408318221638043351819504715129916947301658627} a^{18} - \frac{15041012101629999539507197095981767345567213136687947422601901103304583513}{231935851073023771121947907278816636443276086703639009430259833894603317254} a^{17} + \frac{57731645828381131187043638342029583625098968048121719328580951701816187267}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{16} - \frac{18842023462514141393925317923248549437036654384702593692769131647619392031}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{15} + \frac{67169564258658529432122671384413628682122263526561943343147964171817241315}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{14} - \frac{24137261647791367012041078507805612078643150280184523636403898866504368661}{115967925536511885560973953639408318221638043351819504715129916947301658627} a^{13} - \frac{20666245570847565778210580204236158009083340408642585695674911814058394727}{115967925536511885560973953639408318221638043351819504715129916947301658627} a^{12} - \frac{73214982018750053780906542353729426926837082969336015537042213332995627541}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{11} - \frac{18876757921825849426920324175123408211244216795936350354294062167638315698}{115967925536511885560973953639408318221638043351819504715129916947301658627} a^{10} + \frac{221103264373645584593175571920847646550966952099436811050889987099069297905}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{9} - \frac{77873285789417196360682028304766572107362633979615166243499460963100051215}{463871702146047542243895814557633272886552173407278018860519667789206634508} a^{8} + \frac{42170067244289577924730066581919904482934653039509587316125261888466990169}{115967925536511885560973953639408318221638043351819504715129916947301658627} a^{7} - \frac{29584581492571445721692280747431693881329187374018894018240562015461386}{200984273026883683814512917919251851337327631458959280268856008574179651} a^{6} + \frac{54363495461038230266000344139080860280683746734364141708685178099351012149}{231935851073023771121947907278816636443276086703639009430259833894603317254} a^{5} + \frac{13206270554908585168034382023699953749587629098561019699245894525581796111}{231935851073023771121947907278816636443276086703639009430259833894603317254} a^{4} - \frac{83008866912629297934034048404259068344808481406804178962528449735453038895}{231935851073023771121947907278816636443276086703639009430259833894603317254} a^{3} + \frac{44860669829172740590700821892588708877165757318010668831253686971374045441}{231935851073023771121947907278816636443276086703639009430259833894603317254} a^{2} + \frac{30591334773161125663229074105359209532648144748178739159206365934133992662}{115967925536511885560973953639408318221638043351819504715129916947301658627} a + \frac{18426697688504445930452253594505084850188830551712820756267035331545443916}{115967925536511885560973953639408318221638043351819504715129916947301658627}$

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:  $9$
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:  \( 3049153087.65 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.12.26.87$x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{8} + 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
97Data not computed
2657Data not computed