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Properties

Label 20.0.17930108780...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}$
Root discriminant $57.90$
Ramified primes $2, 3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![358394404, -524324336, 286416976, -22835628, -57380232, 42851868, -15950598, 812556, 2141184, -1108412, 376063, -94802, -513, 9012, -3102, 1104, -144, -48, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 13*x^18 - 48*x^17 - 144*x^16 + 1104*x^15 - 3102*x^14 + 9012*x^13 - 513*x^12 - 94802*x^11 + 376063*x^10 - 1108412*x^9 + 2141184*x^8 + 812556*x^7 - 15950598*x^6 + 42851868*x^5 - 57380232*x^4 - 22835628*x^3 + 286416976*x^2 - 524324336*x + 358394404)
 
gp: K = bnfinit(x^20 - 2*x^19 + 13*x^18 - 48*x^17 - 144*x^16 + 1104*x^15 - 3102*x^14 + 9012*x^13 - 513*x^12 - 94802*x^11 + 376063*x^10 - 1108412*x^9 + 2141184*x^8 + 812556*x^7 - 15950598*x^6 + 42851868*x^5 - 57380232*x^4 - 22835628*x^3 + 286416976*x^2 - 524324336*x + 358394404, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 13 x^{18} - 48 x^{17} - 144 x^{16} + 1104 x^{15} - 3102 x^{14} + 9012 x^{13} - 513 x^{12} - 94802 x^{11} + 376063 x^{10} - 1108412 x^{9} + 2141184 x^{8} + 812556 x^{7} - 15950598 x^{6} + 42851868 x^{5} - 57380232 x^{4} - 22835628 x^{3} + 286416976 x^{2} - 524324336 x + 358394404 \)

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:  \(179301087802125125222400000000000000=2^{28}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.90$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{9} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3}$, $\frac{1}{6} a^{15} + \frac{1}{3} a^{9} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{6} a^{16} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3}$, $\frac{1}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{19} - \frac{1468855230481799274680908170866498254175280566422844397000986595806061425}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{18} - \frac{12646129727610253068371580216485359050559740364420174886231117623523185501}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{17} + \frac{8616571113638610540507856837337115813601236617591107823341815684648167285}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{16} + \frac{7530446038600295146328468655015824119568574377288893541839264007749609145}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{15} + \frac{4980687253916107375436312989140343311076370544428255801339519091705642293}{89938667022635239480773587194011524285618886513816096201175497682798472271} a^{14} + \frac{1057691513993248209971744924869121527072913768235414665549664656989258581}{59959111348423492987182391462674349523745924342544064134116998455198981514} a^{13} - \frac{1511307735743015807575423251591530548224475316132213428017892522633465549}{59959111348423492987182391462674349523745924342544064134116998455198981514} a^{12} + \frac{11324541901455214509607785414658613692621463687989836618152895800138857759}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{11} - \frac{13985569635609611133632851574725248300565576887446343293621070188608329445}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{10} + \frac{40663505121832291154029905677608263986157412464887990689141386817049104226}{89938667022635239480773587194011524285618886513816096201175497682798472271} a^{9} + \frac{37564039798497772171609084049877684512545230045416542529408073670797646910}{89938667022635239480773587194011524285618886513816096201175497682798472271} a^{8} - \frac{35850382889564123571388720497564267670628358830426404529408508885291864167}{179877334045270478961547174388023048571237773027632192402350995365596944542} a^{7} + \frac{25915172782618235766585614303920895118677869549606373955772902388159787426}{89938667022635239480773587194011524285618886513816096201175497682798472271} a^{6} - \frac{4271410041348620635531723094923113424143139753274435993288513515221161078}{89938667022635239480773587194011524285618886513816096201175497682798472271} a^{5} + \frac{343832735089151144076526683853947271861684885781081950694319245109180625}{29979555674211746493591195731337174761872962171272032067058499227599490757} a^{4} + \frac{7567580037405303672876487826585183597316360707120159215661884505433635603}{29979555674211746493591195731337174761872962171272032067058499227599490757} a^{3} + \frac{39515906521342688541078887818528295174222305503153003757715941235770067071}{89938667022635239480773587194011524285618886513816096201175497682798472271} a^{2} - \frac{7131113080501317680074682857481737467047134617735141918409521982379795097}{29979555674211746493591195731337174761872962171272032067058499227599490757} a - \frac{41436833249315184842671067039359006691383630730994055997060250099256370017}{89938667022635239480773587194011524285618886513816096201175497682798472271}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 1949267111.5946145 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-15}, \sqrt{-21})\), 5.1.162000.1, 10.2.141146530560000000.1, 10.0.393660000000.1, 10.0.84687918336000000.1

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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$