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Properties

Label 20.0.35687502577...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{34}\cdot 19^{10}$
Root discriminant $67.24$
Ramified primes $5, 19$
Class number $75020$ (GRH)
Class group $[11, 6820]$ (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![190774501, -158171460, 232462355, -159513940, 134373945, -77459025, 48478500, -23857065, 12113975, -5146960, 2196472, -807385, 292905, -92265, 28295, -7450, 1875, -390, 75, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1875*x^16 - 7450*x^15 + 28295*x^14 - 92265*x^13 + 292905*x^12 - 807385*x^11 + 2196472*x^10 - 5146960*x^9 + 12113975*x^8 - 23857065*x^7 + 48478500*x^6 - 77459025*x^5 + 134373945*x^4 - 159513940*x^3 + 232462355*x^2 - 158171460*x + 190774501)
 
gp: K = bnfinit(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1875*x^16 - 7450*x^15 + 28295*x^14 - 92265*x^13 + 292905*x^12 - 807385*x^11 + 2196472*x^10 - 5146960*x^9 + 12113975*x^8 - 23857065*x^7 + 48478500*x^6 - 77459025*x^5 + 134373945*x^4 - 159513940*x^3 + 232462355*x^2 - 158171460*x + 190774501, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 75 x^{18} - 390 x^{17} + 1875 x^{16} - 7450 x^{15} + 28295 x^{14} - 92265 x^{13} + 292905 x^{12} - 807385 x^{11} + 2196472 x^{10} - 5146960 x^{9} + 12113975 x^{8} - 23857065 x^{7} + 48478500 x^{6} - 77459025 x^{5} + 134373945 x^{4} - 159513940 x^{3} + 232462355 x^{2} - 158171460 x + 190774501 \)

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:  \(3568750257720821537077426910400390625=5^{34}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.24$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(475=5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{475}(1,·)$, $\chi_{475}(324,·)$, $\chi_{475}(134,·)$, $\chi_{475}(341,·)$, $\chi_{475}(151,·)$, $\chi_{475}(474,·)$, $\chi_{475}(284,·)$, $\chi_{475}(94,·)$, $\chi_{475}(96,·)$, $\chi_{475}(419,·)$, $\chi_{475}(229,·)$, $\chi_{475}(39,·)$, $\chi_{475}(189,·)$, $\chi_{475}(436,·)$, $\chi_{475}(286,·)$, $\chi_{475}(246,·)$, $\chi_{475}(56,·)$, $\chi_{475}(379,·)$, $\chi_{475}(381,·)$, $\chi_{475}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{20956633553867130314849059848437738815189306731669624097844149} a^{19} + \frac{10316081396593987457357259827234711855803961978310847524431674}{20956633553867130314849059848437738815189306731669624097844149} a^{18} - \frac{5385658786725794480047771916313040918715586150138332101763597}{20956633553867130314849059848437738815189306731669624097844149} a^{17} + \frac{8070814778917631665223761530217828024719888962162368004643830}{20956633553867130314849059848437738815189306731669624097844149} a^{16} - \frac{2406158194385316852830931327364088445429039350411378959763800}{20956633553867130314849059848437738815189306731669624097844149} a^{15} - \frac{329203922095578984472842661755773318635225496764450361297098}{20956633553867130314849059848437738815189306731669624097844149} a^{14} + \frac{8545410684045281810633230697814101035110293268588370280006925}{20956633553867130314849059848437738815189306731669624097844149} a^{13} + \frac{10141406653452721164743025718521481830789686056173993052739181}{20956633553867130314849059848437738815189306731669624097844149} a^{12} - \frac{5853561297447580199977293616800953514710390358797819574081072}{20956633553867130314849059848437738815189306731669624097844149} a^{11} + \frac{9305481079083423332819462702091296820093993817862899000934124}{20956633553867130314849059848437738815189306731669624097844149} a^{10} - \frac{4482570532051483692289295264899000465972740435938019414199648}{20956633553867130314849059848437738815189306731669624097844149} a^{9} - \frac{4739520478687461415841652932149809349045771721170893892304987}{20956633553867130314849059848437738815189306731669624097844149} a^{8} - \frac{9146081872303256774136983269031503606681052562275529558388415}{20956633553867130314849059848437738815189306731669624097844149} a^{7} - \frac{3118634891054928715369542911851622030329346235935297715805041}{20956633553867130314849059848437738815189306731669624097844149} a^{6} - \frac{2315179470047081255430497253124368041557428696687958057336325}{20956633553867130314849059848437738815189306731669624097844149} a^{5} - \frac{3543939664224829922080652055112823873528033246508460220638349}{20956633553867130314849059848437738815189306731669624097844149} a^{4} + \frac{4653378230619741116181711028991345265452188825882243812862362}{20956633553867130314849059848437738815189306731669624097844149} a^{3} - \frac{8771758297941805133007435221652413777166906895002601410914712}{20956633553867130314849059848437738815189306731669624097844149} a^{2} - \frac{7044228332121201694069743829826163510151517725958884818245968}{20956633553867130314849059848437738815189306731669624097844149} a - \frac{3539436754242449531045370566004053118910047837713648027169241}{20956633553867130314849059848437738815189306731669624097844149}$

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

Class group and class number

$C_{11}\times C_{6820}$, which has order $75020$ (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:  \( 161406.837641 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}, \sqrt{-19})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.1889113616943359375.3, 10.0.377822723388671875.3

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.1$x^{10} - 5 x^{8} + 5$$10$$1$$17$$C_{10}$$[2]_{2}$
$19$19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$