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Properties

Label 20.0.40425381270...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{16}$
Root discriminant $302.23$
Ramified primes $2, 3, 5, 41$
Class number $109827520$ (GRH)
Class group $[2, 2, 2, 2, 2, 3432110]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![177674855049, 29389823568, 78239489544, 10059206184, 15346327698, 1132575984, 1855179564, 43372644, 161496439, -557284, 11227130, -288780, 572590, -33104, 20818, -1220, 946, -48, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 - 48*x^17 + 946*x^16 - 1220*x^15 + 20818*x^14 - 33104*x^13 + 572590*x^12 - 288780*x^11 + 11227130*x^10 - 557284*x^9 + 161496439*x^8 + 43372644*x^7 + 1855179564*x^6 + 1132575984*x^5 + 15346327698*x^4 + 10059206184*x^3 + 78239489544*x^2 + 29389823568*x + 177674855049)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 - 48*x^17 + 946*x^16 - 1220*x^15 + 20818*x^14 - 33104*x^13 + 572590*x^12 - 288780*x^11 + 11227130*x^10 - 557284*x^9 + 161496439*x^8 + 43372644*x^7 + 1855179564*x^6 + 1132575984*x^5 + 15346327698*x^4 + 10059206184*x^3 + 78239489544*x^2 + 29389823568*x + 177674855049, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} - 48 x^{17} + 946 x^{16} - 1220 x^{15} + 20818 x^{14} - 33104 x^{13} + 572590 x^{12} - 288780 x^{11} + 11227130 x^{10} - 557284 x^{9} + 161496439 x^{8} + 43372644 x^{7} + 1855179564 x^{6} + 1132575984 x^{5} + 15346327698 x^{4} + 10059206184 x^{3} + 78239489544 x^{2} + 29389823568 x + 177674855049 \)

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:  \(40425381270649334011516273744683372380160000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $302.23$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(2051,·)$, $\chi_{4920}(1349,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(2189,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(3011,·)$, $\chi_{4920}(2839,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(1691,·)$, $\chi_{4920}(1759,·)$, $\chi_{4920}(3749,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(611,·)$, $\chi_{4920}(2789,·)$, $\chi_{4920}(2599,·)$, $\chi_{4920}(1451,·)$, $\chi_{4920}(4159,·)$, $\chi_{4920}(2429,·)$, $\chi_{4920}(3199,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{1}{9} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{10}{27} a^{5} - \frac{10}{27} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{1}{27} a^{10} + \frac{2}{27} a^{8} - \frac{1}{27} a^{6} + \frac{1}{3} a^{5} - \frac{1}{27} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{15} + \frac{1}{81} a^{14} - \frac{4}{81} a^{12} + \frac{2}{81} a^{11} + \frac{1}{27} a^{10} - \frac{4}{81} a^{9} - \frac{10}{81} a^{8} - \frac{1}{9} a^{7} + \frac{4}{81} a^{6} + \frac{7}{81} a^{5} - \frac{7}{27} a^{4} - \frac{8}{27} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{81} a^{16} - \frac{1}{81} a^{14} - \frac{1}{81} a^{13} + \frac{1}{27} a^{12} - \frac{2}{81} a^{11} - \frac{4}{81} a^{10} + \frac{1}{27} a^{9} + \frac{10}{81} a^{8} + \frac{10}{81} a^{7} + \frac{2}{27} a^{6} - \frac{25}{81} a^{5} - \frac{11}{27} a^{4} + \frac{5}{27} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{243} a^{17} - \frac{1}{243} a^{15} + \frac{2}{243} a^{14} + \frac{1}{81} a^{13} + \frac{1}{243} a^{12} + \frac{5}{243} a^{11} - \frac{1}{81} a^{10} + \frac{1}{243} a^{9} + \frac{16}{243} a^{8} - \frac{7}{81} a^{7} - \frac{1}{243} a^{6} - \frac{32}{81} a^{5} - \frac{11}{81} a^{4} - \frac{1}{3} a^{3} - \frac{10}{27} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{912086487483410936657511} a^{18} - \frac{1234088181987709418507}{912086487483410936657511} a^{17} - \frac{4134608205312420487180}{912086487483410936657511} a^{16} - \frac{24861834290975614751}{12494335444978232009007} a^{15} - \frac{11061408586440150007594}{912086487483410936657511} a^{14} - \frac{16195614033799502348882}{912086487483410936657511} a^{13} - \frac{4290958538385393450127}{101342943053712326295279} a^{12} - \frac{43836556820050356503776}{912086487483410936657511} a^{11} + \frac{16805635219165770279628}{912086487483410936657511} a^{10} - \frac{42483126029750430233587}{912086487483410936657511} a^{9} - \frac{23393783357219585208926}{912086487483410936657511} a^{8} - \frac{124860429727531087684126}{912086487483410936657511} a^{7} - \frac{129186933376209856515118}{912086487483410936657511} a^{6} + \frac{5239215723514352943668}{11260327005968036255031} a^{5} - \frac{103079974486709833770983}{304028829161136978885837} a^{4} + \frac{15026380828862001051217}{33780981017904108765093} a^{3} + \frac{37829691526417824246176}{101342943053712326295279} a^{2} + \frac{998186761119700877807}{33780981017904108765093} a - \frac{1472281836341168851873}{11260327005968036255031}$, $\frac{1}{12469701946482477980826830883347643672496783357610835693} a^{19} - \frac{511819983104902091770602497482}{12469701946482477980826830883347643672496783357610835693} a^{18} + \frac{9958213795844903217746241082698739146161686348117907}{12469701946482477980826830883347643672496783357610835693} a^{17} + \frac{19686537827204295116542298119686010784079677286353515}{4156567315494159326942276961115881224165594452536945231} a^{16} - \frac{53760338357174107706617849254855744458252441742107816}{12469701946482477980826830883347643672496783357610835693} a^{15} + \frac{113603588401089418805236280781691104326940575904072595}{12469701946482477980826830883347643672496783357610835693} a^{14} + \frac{161098502492854405865895267648098006620259003737266613}{12469701946482477980826830883347643672496783357610835693} a^{13} - \frac{242387123003626563097135761769269262135334889895827701}{12469701946482477980826830883347643672496783357610835693} a^{12} + \frac{336449742891422828744692127779671762524768207808791524}{12469701946482477980826830883347643672496783357610835693} a^{11} + \frac{28686425223322520412598139178919603665869101827412592}{4156567315494159326942276961115881224165594452536945231} a^{10} - \frac{415793472516063301154735463641402651291798388673435279}{12469701946482477980826830883347643672496783357610835693} a^{9} - \frac{1840283643008668647385453763532665132725596645203880541}{12469701946482477980826830883347643672496783357610835693} a^{8} - \frac{966069559357513220876361355526181718091650728575271080}{12469701946482477980826830883347643672496783357610835693} a^{7} - \frac{156331295735594535370930276106076352528942869737398052}{1385522438498053108980758987038627074721864817512315077} a^{6} + \frac{1189321384494196544715866133434243242152541284665335985}{4156567315494159326942276961115881224165594452536945231} a^{5} - \frac{241678797407430267307332788582101847139320889518202362}{1385522438498053108980758987038627074721864817512315077} a^{4} + \frac{64159967254136101606607181890392214353709493492694672}{1385522438498053108980758987038627074721864817512315077} a^{3} - \frac{12896092089457292277958730106799045473366503969448469}{51315645870298263295583666186615817582291289537493151} a^{2} + \frac{4681866862823674069768411662348618416433014897119882}{51315645870298263295583666186615817582291289537493151} a + \frac{14557548627790962217654617282936118892027154439982801}{51315645870298263295583666186615817582291289537493151}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{3432110}$, which has order $109827520$ (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:  \( 9452086796.91891 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-5}, \sqrt{6})\), 5.5.2825761.1, 10.0.25551760733187200000.2, 10.10.63580957267604373504.1, 10.0.198690491461263667200000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$