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Properties

Label 20.0.55039620640...7152.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 17^{10}\cdot 53^{13}$
Root discriminant $108.90$
Ramified primes $2, 17, 53$
Class number $880$ (GRH)
Class group $[2, 2, 2, 110]$ (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87491689, -104041302, 226406567, -96683974, 120949292, 11751366, 13861755, 22409014, 487944, 4710184, 1013178, 356230, 237865, 13740, 25166, -638, 1787, -128, 72, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 72*x^18 - 128*x^17 + 1787*x^16 - 638*x^15 + 25166*x^14 + 13740*x^13 + 237865*x^12 + 356230*x^11 + 1013178*x^10 + 4710184*x^9 + 487944*x^8 + 22409014*x^7 + 13861755*x^6 + 11751366*x^5 + 120949292*x^4 - 96683974*x^3 + 226406567*x^2 - 104041302*x + 87491689)
 
gp: K = bnfinit(x^20 - 4*x^19 + 72*x^18 - 128*x^17 + 1787*x^16 - 638*x^15 + 25166*x^14 + 13740*x^13 + 237865*x^12 + 356230*x^11 + 1013178*x^10 + 4710184*x^9 + 487944*x^8 + 22409014*x^7 + 13861755*x^6 + 11751366*x^5 + 120949292*x^4 - 96683974*x^3 + 226406567*x^2 - 104041302*x + 87491689, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 72 x^{18} - 128 x^{17} + 1787 x^{16} - 638 x^{15} + 25166 x^{14} + 13740 x^{13} + 237865 x^{12} + 356230 x^{11} + 1013178 x^{10} + 4710184 x^{9} + 487944 x^{8} + 22409014 x^{7} + 13861755 x^{6} + 11751366 x^{5} + 120949292 x^{4} - 96683974 x^{3} + 226406567 x^{2} - 104041302 x + 87491689 \)

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:  \(55039620640627142810521394077176301617152=2^{20}\cdot 17^{10}\cdot 53^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.90$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{465604988736997145724761155617486559687198167524008779259995912550999007547892} a^{19} - \frac{6793308276831053791595996168782023917431392063475408910226229344554745193217}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{18} + \frac{87005768263980494645738869929249877482138748229851908630926132572901121611635}{465604988736997145724761155617486559687198167524008779259995912550999007547892} a^{17} - \frac{109370787885866632393252295490695606066054203711435795491889122001552861390203}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{16} - \frac{8732380694486108366107528663670348468696626152708070768780831727697287467190}{116401247184249286431190288904371639921799541881002194814998978137749751886973} a^{15} - \frac{20355152467394957079077933349513936411127054963693711428928674415782656787033}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{14} + \frac{113511609430254797599900424500407068186865757753395994691777245000323873764263}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{13} + \frac{43460136578282435948863263541636265247286784271409397972989603722737947708869}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{12} + \frac{221631592134088976740536502791211914363032173405328425006158411570340214107807}{465604988736997145724761155617486559687198167524008779259995912550999007547892} a^{11} + \frac{82626349598829825305803382011237052841885875146235647446205294568144378010385}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{10} + \frac{93692257758636742688332842649623246925330899292278821658586183309247101166899}{465604988736997145724761155617486559687198167524008779259995912550999007547892} a^{9} - \frac{42212017881813954674144352930698125047225089916823239838653074078461630376961}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{8} - \frac{227016868843046253136195672259046851219456473738485021124100169819993599502127}{465604988736997145724761155617486559687198167524008779259995912550999007547892} a^{7} + \frac{10375999943275140432387667682154529703558403325796370702403607872379368675871}{116401247184249286431190288904371639921799541881002194814998978137749751886973} a^{6} - \frac{111186093815833745036230896609693848006042721068316571367544036765799557638581}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{5} - \frac{42747645468254031656811531173717020871252100067143287638251478359255689679721}{116401247184249286431190288904371639921799541881002194814998978137749751886973} a^{4} + \frac{42053407163035770241603118750151192907249678988587175522761550896393203770619}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{3} - \frac{113274140016597380922863388192307063176013076913131765933256858718492718435497}{232802494368498572862380577808743279843599083762004389629997956275499503773946} a^{2} - \frac{101241748393363295861865526028611481669113935789170833204332758754580123988191}{465604988736997145724761155617486559687198167524008779259995912550999007547892} a + \frac{55771897979092808962974965231227979657900308860155955942820316526936564860549}{232802494368498572862380577808743279843599083762004389629997956275499503773946}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{110}$, which has order $880$ (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:  \( 4097659676.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.245072.1, 5.5.2382032.1, 10.10.8056377164681869568.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$